Apomorphisms in TypeScript: Unpacking and Reconstructing Data Structures

Apomorphisms are a powerful generalization of catamorphisms (folds) and anamorphisms (unfolds) that allow for both deconstruction and reconstruction of data structures. This challenge focuses on implementing apomorphisms for common recursive data structures in TypeScript, enabling flexible and type-safe operations on them.

Problem Description

You are tasked with implementing a generic apomorphism function for recursive data structures in TypeScript. An apomorphism is a recursive function that can be thought of as a generalized foldMap that also handles the "unfolding" or construction of the data structure.

Specifically, you need to:

1. Define a generic Apomorphism type: This type should represent the core of the apomorphic operation. It will take two type parameters: T (the target recursive data structure) and B (the accumulated result type). The Apomorphism type should have a method to deconstruct T into a base case or a structure that can be recursively processed, and a method to reconstruct T from a base case and the results of recursive calls.

2. Implement an apomorphism function: This function will accept an Apomorphism and a value of the target data structure T, and return a value of type B. The function should recursively apply the deconstruction and reconstruction logic defined in the Apomorphism.

3. Apply to common recursive structures: Demonstrate the use of your apomorphism implementation with at least two common recursive data structures, such as:

   • A binary tree
   • A list

Key Requirements

• Type Safety: All implementations must be fully type-safe in TypeScript.
• Genericity: The apomorphism function and the Apomorphism type should be generic enough to work with different recursive data structures and result types.
• Clear Deconstruction and Reconstruction: The Apomorphism type should clearly separate the logic for taking apart the structure and putting it back together.

Expected Behavior

The apomorphism function should correctly traverse the input data structure, applying the deconstruction logic and then using the reconstruction logic to build the final result.

Edge Cases

• Empty data structures (e.g., an empty list, a null tree).
• Data structures with only one node/element.

Examples

Let's consider a simple binary tree structure first.

// Hypothetical Tree definition for illustration
interface EmptyTree<A> {
  type: 'empty';
}

interface Node<A> {
  type: 'node';
  value: A;
  left: Tree<A>;
  right: Tree<A>;
}

type Tree<A> = EmptyTree<A> | Node<A>;

// Example Tree:
//      5
//     / \
//    3   8
//   /
//  1

const exampleTree: Tree<number> = {
  type: 'node',
  value: 5,
  left: {
    type: 'node',
    value: 3,
    left: {
      type: 'node',
      value: 1,
      left: { type: 'empty' },
      right: { type: 'empty' },
    },
    right: { type: 'empty' },
  },
  right: {
    type: 'node',
    value: 8,
    left: { type: 'empty' },
    right: { type: 'empty' },
  },
};

Example 1: Sum of Tree Elements

Input:

// Apomorphism to sum elements
const sumApomorphism: Apomorphism<Tree<number>, number> = {
  // Deconstruction:
  // If it's an empty tree, return 0 (the base case for sum).
  // If it's a node, return its value and recursively process left and right children.
  cata: (tree) => {
    if (tree.type === 'empty') {
      return 0; // Base case for summation
    } else {
      // For nodes, we need to return a structure that the ana part can use.
      // This structure typically mirrors the original recursive step.
      // Here, we return the node's value and references to its children,
      // which will be recursively processed.
      return {
        value: tree.value,
        left: tree.left,
        right: tree.right,
      };
    }
  },
  // Reconstruction:
  // The 'ana' function takes a 'seed' and a function to produce the next step.
  // Here, the 'seed' will be the result of the cata step.
  // The function passed to 'ana' (let's call it 'step') will take the 'seed'
  // and produce either a fully constructed structure (if it's a base case)
  // or the next intermediate structure to be processed.
  //
  // For a sum, the 'ana' part isn't directly "reconstructing" the tree in the same way,
  // but it defines how the recursive calls are "built up".
  // In a true apomorphism, 'cata' deconstructs and 'ana' reconstructs.
  //
  // For this exercise, we'll focus on a common interpretation where 'cata'
  // extracts the recursive structure and 'ana' uses a "seed" to guide construction.
  // For a summation, the 'cata' step extracts the values and structure,
  // and the 'ana' step would be implicitly handled by the recursive traversal.
  //
  // A more direct apomorphism requires both a deconstruction function AND
  // a reconstruction function that takes a 'seed' and returns a recursive structure.
  //
  // Let's refine the Apomorphism definition for clarity:
  //
  // type Apomorphism<T, B, S> = {
  //   // Takes T and returns a 'step' structure or a base result.
  //   // 'S' is the 'seed' or intermediate state type.
  //   cata: (t: T) => T | B; // Simplified: returns a recursive step or base result
  //   // Takes a 'seed' (S) and produces the next T (or base case).
  //   ana: (s: S) => T | B;
  // };
  //
  // This is getting complex. For this problem, we'll focus on a more practical
  // interpretation of apomorphism for common use cases like `foldMap` and `unfoldr`.
  //
  // Let's define Apomorphism as a pair of functions that work together:
  // 1. A way to "embed" a value into the recursive structure (analogous to `pure` in Monads or `inject` in Functors).
  // 2. A way to "extract" the recursive structure (analogous to `project` in Functors).
  //
  // A more direct way to think about it for this challenge is:
  // Define a type that describes how to process *each step* of the recursion.
  // This step involves:
  // a) What is the base case for this structure and result type?
  // b) Given an element/node and the results from its children, how do we combine them?
  // c) Given a "seed", how do we produce the next element/node or a base case?
  //
  // Let's redefine Apomorphism to align with common functional programming patterns:
  //
  // Apomorphism<F<_>, A> = {
  //   embed: <B>(fa: F<B>) => F<A>; // For constructing from inner structure
  //   project: <B>(fa: F<A>) => F<B> | A; // For deconstructing into inner structure or base
  // }
  //
  // This is still abstract. For this challenge, let's simplify and focus on the
  // `cata` (deconstruction) and `ana` (reconstruction) pair that directly operates on the structure.
  //
  // Let's reconsider the `Apomorphism` type:
  //
  // type Apomorphism<T_Rec, B> = {
  //   // `cata` maps the recursive structure `T_Rec` to an intermediate representation `F<S>`
  //   // where `S` is the type of the recursive step's result, and `F` is a functor.
  //   // For simplicity, let's assume `F` can be represented by the structure itself or a simpler form.
  //   // This is where the "unpacking" happens.
  //   cata: (t: T_Rec) => any; // Returns a value that the 'ana' can process, or a base value.
  //
  //   // `ana` maps a "seed" `S` to the recursive structure `T_Rec`.
  //   // This is where the "reconstruction" happens.
  //   ana: (seed: any) => T_Rec;
  // };
  //
  // This is still too abstract for a direct TS implementation without more context on `F`.
  //
  // Let's adopt a pragmatic approach by defining an `apomorphism` function that takes
  // two functions: `cataStep` and `anaStep`.
  //
  // `cataStep`: Deconstructs a node into its components and recursive calls (or base case).
  // `anaStep`: Reconstructs a node from its components and recursive results (or base case).
  //
  // The `apomorphism` function will then use these to perform the recursive traversal.
  //
  // **Revised Approach for the Challenge:**
  //
  // We will define an `Apomorphism` as a structure that encapsulates the logic for
  // processing each level of recursion. It will have:
  //
  // 1.  A `cata` function: This function deconstructs a value of the recursive type `T`
  //     into a structure that can be further processed. It returns either a base result
  //     or a structure containing the components and recursive calls.
  // 2.  An `ana` function: This function takes a "seed" value and reconstructs a value
  //     of type `T`, potentially from the results of recursive computations.
  //
  // This is still a bit fuzzy. Let's anchor this to a more concrete pattern seen in libraries.
  //
  // The concept of apomorphism often arises in the context of *traversable* data structures.
  // A common pattern is `traverse` which uses an `Applicative`.
  //
  // For this challenge, let's focus on the structure of `cata` and `ana` working together.
  //
  // `cata`: Takes a `T` and returns a structure representing its components and recursive structure.
  // `ana`: Takes a "seed" and constructs a `T`.
  //
  // Let's simplify the `Apomorphism` type for this specific exercise:
  //
  // `Apomorphism<T, B>`: where `T` is the recursive type and `B` is the result type.
  //
  // The core idea is that `cata` unpacks `T` into a form that `ana` can use to rebuild `T` (or a related structure).
  //
  // For a structure like `Tree<A>`, `cata` might map `Tree<A>` to `Option<{ value: A, left: Tree<A>, right: Tree<A> }>`.
  // And `ana` might map `Option<{ value: A, left: Tree<A>, right: Tree<A> }>` back to `Tree<A>`.
  //
  // This problem is about generalizing this pair of functions.

  // **Let's define the `Apomorphism` type and the `apomorphism` function first, then apply it.**

  // We need a way to represent the "recursive step" for both deconstruction and reconstruction.
  // Let's call this the `Projected` type.
  // `Projected<T_Rec, B_Inner>`: Represents the deconstructed form.
  // `Injected<T_Rec, B_Inner>`: Represents the reconstructed form.

  // This is leaning towards Free Monads or similar advanced structures.
  // Let's simplify to a more direct recursive pattern:

  // **Focus on the combined deconstruction and reconstruction:**
  // An apomorphism can be seen as a function that maps a recursive structure `F<A>`
  // to a structure `G<B>` where `G` can be seen as the "base case" or the "recursive step result".

  // Let's try a concrete example first and then generalize.

  // **For summing a tree:**
  // cata(tree):
  //   - If empty, return `0` (base value).
  //   - If node(v, l, r), return `{ value: v, left: cata(l), right: cata(r) }`.
  //
  // This `cata` is more like a catamorphism. An apomorphism needs an `ana` part.
  //
  // Let's assume our `Apomorphism` type takes the recursive structure and the final result type.
  //
  // `Apomorphism<T_Rec, B>`:
  //   - `cata`: `(t: T_Rec) => SomeDeconstructedForm | B`
  //   - `ana`: `(seed: SomeSeed) => T_Rec | B`
  //
  // The `apomorphism` function will take `Apomorphism<T_Rec, B>` and `T_Rec`.

  // Let's refine the problem statement to be more direct about the structure of the `Apomorphism`.
  // It should define how to:
  // 1. "Project" the recursive structure into a non-recursive form plus recursive calls.
  // 2. "Inject" a non-recursive form back into the recursive structure.

  // For a binary tree `Tree<A>`:
  // Projection: `Tree<A> -> Option<{ value: A, left: Tree<A>, right: Tree<A> }>`
  // Injection: `Option<{ value: A, left: Tree<A>, right: Tree<A> }> -> Tree<A>`

  // This is still abstract. Let's simplify the `Apomorphism` definition to be practical for TS.

  // **Revised Apomorphism Definition for this Challenge:**
  // We define an `Apomorphism` for a recursive type `T` and a result type `B`.
  // It consists of two functions:
  //
  // 1. `project`: Takes a value of type `T` and returns either a base result of type `B`
  //    or an intermediate structure that contains the "recursive components" of `T`
  //    and the recursive calls. The "recursive components" are the parts of `T` that
  //    are not recursive themselves (e.g., `value` in a `Node`), and the "recursive calls"
  //    are placeholders for the results from processing the children.
  //
  // 2. `embed`: Takes an intermediate structure (of the same shape as the one returned
  //    by `project` for a recursive case) and reconstructs a value of type `T`.
  //    This `embed` function will also take the results of the recursive computations
  //    that correspond to its "recursive calls".
  //
  // The `apomorphism` function will orchestrate these.

  // Let's re-evaluate Example 1 with this refined understanding.

  // **Example 1: Sum of Tree Elements (Revised)**
  //
  // We need to define an `Apomorphism` for `Tree<number>` that results in `number`.
  //
  // The `project` function should deconstruct the tree.
  // The `embed` function should reconstruct it (or combine results).
  //
  // For summation, the core operation is combining results.
  //
  // `project: Tree<number> -> { value: number, left: Tree<number>, right: Tree<number> } | number`
  //   - If `EmptyTree`, return `0`.
  //   - If `Node(v, l, r)`, return `{ value: v, left: l, right: r }`. (This is the deconstruction part)
  //
  // `embed: (intermediate: { value: number, left: Tree<number>, right: Tree<number> }, sum_left: number, sum_right: number) => number`
  //   - This `embed` function would take the decomposed parts *and the results of recursive calls*.
  //   - For sum: `embed({ value: v, left: l, right: r }, sum_l, sum_r) => v + sum_l + sum_r`
  //
  // This implies our `apomorphism` function will recursively call itself.

  // **Let's try to define the `Apomorphism` type and `apomorphism` function first.**

  // The `project` function needs to capture the recursive structure.
  // Let `F<A>` be the "recursive functor" for our structure (e.g., for `Tree<A>`, `F<A>` could be `Tree<A>` itself, or a simplified version for the recursion).
  //
  // `project: T -> BaseResult | F<T>` (where `F<T>` represents the recursive structure to be processed further)
  // `embed: F<B> -> T` (where `B` is the result type from recursive calls, and `F<B>` represents those results combined in the structure)
  //
  // This is still quite abstract. Let's simplify the target.

  // **Challenge Goal: Implement `apomorphism` for a recursive type `T` with a result type `B`.**
  //
  // The `apomorphism` function will take:
  // 1. A "pattern matching" function `cata` that deconstructs `T`.
  //    `cata: (t: T) => BaseCase<B> | RecursiveStep<T_Rec, B_Intermediate>`
  //    Where `BaseCase<B>` is the base result, and `RecursiveStep` describes how to get the next `T` and how to combine results.
  //
  // 2. A "construction" function `ana`.
  //    `ana: (seed: B_Intermediate) => T | BaseCase<B>`
  //
  // This is still too complex. Let's focus on the core idea of apomorphism: combining `cata` and `ana`.
  //
  // A more direct interpretation in terms of mapping:
  // `cata` maps `T` to some representation of its structure for processing.
  // `ana` maps a seed to `T`.
  //
  // **The `apomorphism` function itself will be the recursion.**
  //
  // Let's define an `Apomorphism` object that encapsulates the logic for a specific recursive type and result.

  // **Revised `Apomorphism` Definition:**
  //
  // `Apomorphism<T_Rec, B>`:
  //   - `cata`: `(t: T_Rec) => BaseResult<B> | RecursiveInfo<T_Rec>`
  //     `BaseResult<B>`: A value of type `B` (the base case).
  //     `RecursiveInfo<T_Rec>`: A structure containing the components needed to reconstruct
  //                             the current level and the sub-structures that need recursive processing.
  //                             The shape of `RecursiveInfo` should be consistent.
  //
  //   - `ana`: `(seed: Seed<B_Intermediate>) => T_Rec | BaseResult<B>`
  //     `Seed<B_Intermediate>`: A value that guides the reconstruction. It typically
  //                             contains the results from recursive calls and potentially
  //                             other data to guide the construction.
  //
  // This is still quite abstract. Let's simplify for the challenge.
  //
  // **The essence of apomorphism is to view a recursive structure `T` as a fixed point of a functor `F`.**
  // `T = Fix F`. Apomorphism involves `cata` (mapping `F<A>` to `A`) and `ana` (mapping `A` to `F<A>`).
  //
  // For this challenge, we will abstract away the `Fix` and `Functor` parts and focus on the `cata` and `ana` operations directly on the recursive type.

  // **Simplest Apomorphism Implementation for this Challenge:**
  //
  // We'll define an `Apomorphism` record containing two functions:
  // 1. `project`: Deconstructs a value of the recursive type `T` into either a base case `B`
  //    or a structure that can be used for recursive calls and reconstruction.
  //    The structure returned for the recursive case must contain placeholders for the
  //    recursive results, and the original components of the structure.
  //
  // 2. `embed`: Reconstructs a value of type `T` from the results of recursive calls
  //    and potentially other data.
  //
  // The `apomorphism` function will take an `Apomorphism<T, B>` and a value of type `T`.

  // **Let's reconsider Example 1 with this definition:**
  //
  // `T = Tree<number>`, `B = number`
  //
  // `Apomorphism<Tree<number>, number>`:
  //
  // `project: (tree: Tree<number>) => number | { value: number, left: Tree<number>, right: Tree<number> }`
  //   - For `EmptyTree`: returns `0`.
  //   - For `Node(v, l, r)`: returns `{ value: v, left: l, right: r }`.
  //
  // `embed: (components: { value: number, left_res: number, right_res: number }) => Tree<number>`
  //   - This is tricky. For summation, we are not "reconstructing" the tree itself,
  //     but rather combining results to get a single `number`.
  //
  // **This implies that the `apomorphism` function will perform the recursion, and the `Apomorphism` definition guides *how* each step is processed.**
  //
  // Let's try defining the `apomorphism` function first, assuming a structure for `Apomorphism`.

  // The `apomorphism` function will look something like this:
  //
  // `apomorphism<T, B>(apo: Apomorphism<T, B>, value: T): B { ... }`
  //
  // Inside `apomorphism`, we'll call `apo.project(value)`.
  // If it's a base case `B`, return `B`.
  // If it's a recursive structure, we need to recursively call `apomorphism` on its children,
  // then use `apo.embed` to combine those results.
  //
  // **This means `apo.project` should return something that `apo.embed` can use, and that "something" should contain the recursive sub-problems.**
  //
  // Let's define the intermediate structure type more concretely.
  //
  // For a binary tree, the recursive structure looks like `{ value: A, left: Tree<A>, right: Tree<A> }`.
  //
  // **Revised Apomorphism Definition for Trees:**
  //
  // `Apomorphism<A, B>` for `Tree<A>`:
  //
  // `project: (tree: Tree<A>) => B | { value: A, left: Tree<A>, right: Tree<A> }`
  // `embed: (components: { value: A, left_result: B, right_result: B }) => B`
  //
  // The `apomorphism` function will then use these:
  //
  // `apomorphism<A, B>(apo: Apomorphism<A, B>, tree: Tree<A>): B {`
  //   `const projection = apo.project(tree);`
  //   `if (typeof projection === 'number' && !isNaN(projection)) { // Check if it's the base result `B` (assuming B is number for now)`
  //     `return projection;`
  //   `} else { // It's the recursive structure`
  //     `const { value, left, right } = projection; // Type assertion needed or refine projection return type`
  //     `const left_result = apomorphism(apo, left);`
  //     `const right_result = apomorphism(apo, right);`
  //     `return apo.embed({ value, left_result, right_result });`
  //   `}`
  // `}`
  //
  // This looks like a functional approach. The `Apomorphism` defines how to *operate on each node*.
  //
  // Let's generalize this.

  // **General `Apomorphism` Type and `apomorphism` Function:**

  // `T`: The recursive data structure.
  // `B`: The final result type.
  // `RecInfo<T>`: The type representing the recursive components of `T`.
  //               This should be a generic way to describe the structure of `T` without the recursion.
  //               For a `Tree<A>`, `RecInfo<Tree<A>>` might be `{ value: A, left: Tree<A>, right: Tree<A> }`.
  //               For a `List<A>`, `RecInfo<List<A>>` might be `{ head: A, tail: List<A> }`.
  //               This implies we need a way to extract this non-recursive shape.

  // This is becoming complex due to the need to define `RecInfo`.
  //
  // **Let's simplify the problem definition to focus on the pattern matching and reconstruction of *known* recursive structures.**
  //
  // **Focus on `cata` and `ana` as state transformers.**
  //
  // `cata`: Maps `T -> ResultOrRecursiveStep<T, B_Inner>`
  // `ana`: Maps `Seed<B_Inner> -> T or BaseResult<B>`

  // **Let's re-define the goal:**
  //
  // Implement a generic `apomorphism` function that takes:
  // 1. A `project` function: Deconstructs a recursive value `T` into either a base result `B`
  //    or an intermediate structure containing the "fields" of `T` and *placeholders* for
  //    recursive calls.
  // 2. An `embed` function: Reconstructs a final result `B` from an intermediate structure
  //    that now contains the *results* of recursive calls.
  //
  // The `apomorphism` function will handle the recursion.

  // **For `Tree<number>` and result `number` (sum):**
  //
  // `project: (tree: Tree<number>) => number | { value: number, left: Tree<number>, right: Tree<number> }`
  //   - `EmptyTree` -> `0`
  //   - `Node(v, l, r)` -> `{ value: v, left: l, right: r }`
  //
  // `embed: (components: { value: number, left_result: number, right_result: number }) => number`
  //   - `{ value, left_result, right_result }` -> `value + left_result + right_result`
  //
  // **The `apomorphism` function:**
  //
  // `apomorphism<T, B>(project: (t: T) => B | Intermediate<T>, embed: (interm_with_results: IntermediateWithResults<B>) => B, value: T): B`
  //
  // This requires defining `Intermediate<T>` and `IntermediateWithResults<B>`.
  // This is still the core difficulty.

  // **Let's make the `Apomorphism` type more explicit about the structure of intermediate steps.**
  //
  // `Apomorphism<T, B, F>` where `F` is the "recursive functor".
  //
  // This is getting too theoretical. Let's simplify.
  //
  // **Problem Reframing: Implement a generic traversal function that supports both deconstruction and reconstruction.**
  //
  // We'll define an `Apomorphism` object that has:
  //
  // 1. `step`: A function that takes a value of the recursive type `T` and returns:
  //    - The final result `B` (if it's a base case).
  //    - Or, a structure that describes the next recursive step, including:
  //      - The original components of the node (e.g., `value`).
  //      - The sub-structures that need to be processed recursively.
  //      - A way to combine the results from the recursive calls.
  //
  // This is essentially what `cata` and `ana` do when combined.

  // **Let's go back to the `cata`/`ana` interpretation, but simplify what they return.**
  //
  // `Apomorphism<T, B>`:
  //   - `cata`: `(t: T) => { base: B } | { recurse: RecursiveStep<T> }`
  //     `RecursiveStep<T>`: Describes how to break down `T` into sub-problems.
  //     For `Tree<A>`: `RecursiveStep<Tree<A>> = { value: A, left: Tree<A>, right: Tree<A> }`
  //
  //   - `ana`: `(seed: AnaSeed<B>) => { base: B } | { recurse: RecursiveBuildStep<AnaSeed<B>> }`
  //     `AnaSeed<B>`: The information needed to reconstruct a piece of the structure. It *must* contain the results from recursive calls.
  //     For `Tree<A>`: `AnaSeed<B> = { value: A, left_result: B, right_result: B }` (assuming `B` is the result type from children)
  //
  // The `apomorphism` function would then:
  //
  // `apomorphism<T, B>(apo: Apomorphism<T, B>, value: T): B {`
  //   `const step = apo.cata(value);`
  //   `if ('base' in step) {`
  //     `return step.base;`
  //   `} else {`
  //     `const { value: nodeValue, left, right } = step.recurse;`
  //     `const left_result = apomorphism(apo, left);`
  //     `const right_result = apomorphism(apo, right);`
  //     `// Now we need to use the 'ana' part to construct the result.`
  //     `// This is where it gets tricky. The 'ana' part typically builds the *structure* T,`
  //     `// not the *result* B directly.`
  //
  // **This suggests that `apomorphism` is not just a single function, but a pattern of recursion.**
  //
  // **Let's define the `Apomorphism` as a combinator for `cata` and `ana`.**
  //
  // `Apomorphism<F<_>, A>`:
  //   - `cata`: `F<A> -> A`
  //   - `ana`: `A -> F<A>`
  //
  // This is the standard definition in functional programming.
  // In TypeScript, `F<_>` needs to be represented.
  //
  // **Let's simplify the goal for this challenge to be implementable in TypeScript without advanced FP libraries:**
  //
  // **Implement `apomorphism` for a specific recursive type `T` and a result type `B`.**
  //
  // The challenge is to create a function `apomorphism` that takes:
  //
  // 1. A `cata_step` function: `(t: T) => B | RecInfo<T>`
  //    - `B`: The base case result.
  //    - `RecInfo<T>`: A structure describing the recursive decomposition. This structure must contain all sub-structures that need to be recursively processed.
  //
  // 2. An `ana_step` function: `(seed: AnaSeed<B>) => T | B`
  //    - `AnaSeed<B>`: A structure that contains the results from recursive calls (`B`) and any other information needed to build the `T` structure.
  //
  // The `apomorphism` function will manage the recursion.

  // **Let's use the `Tree<number>` and sum example again to illustrate `RecInfo` and `AnaSeed`.**
  //
  // `T = Tree<number>`, `B = number`
  //
  // `cata_step: (tree: Tree<number>) => number | { value: number, left: Tree<number>, right: Tree<number> }`
  //   - `EmptyTree` -> `0`
  //   - `Node(v, l, r)` -> `{ value: v, left: l, right: r }`
  //
  // `ana_step: (seed: { value: number, left_result: number, right_result: number }) => Tree<number> | number`
  //   - This `ana_step` needs to be able to *construct* a `Tree<number>` or return a base `number`.
  //   - This is where the apomorphism truly shines: it can *also* construct.
  //   - For the summation example, we are primarily using the `cata` part to get to `B`.
  //   - The `ana` part is implicitly handled by the recursive calls within `apomorphism`.
  //
  // **This implies `apomorphism` function signature should be:**
  //
  // `apomorphism<T, B>(cata: (t: T) => B | RecInfo<T>, ana: (seed: AnaSeed<B>) => T, value: T): B`
  //
  // This definition assumes that `ana` *builds* the structure `T` from seeds.
  //
  // **Let's adjust the problem statement and expected output to reflect this.**
  //
  // The challenge is to implement `apomorphism` for *processing* recursive structures.
  // This processing might result in a new value (`B`) or a reconstructed structure.
  //
  // **Revised Goal:** Implement a generic `apomorphism` function.
  //
  // `apomorphism<T, B>(project: (t: T) => B | RecursiveStep<T>, embed: (recursive_results: RecursiveEmbed<B, T>) => B, value: T): B`
  //
  // `RecursiveStep<T>`: The deconstructed parts of `T` that are themselves recursive.
  // `RecursiveEmbed<B, T>`: The results of recursive calls (`B`) combined with the non-recursive parts of `T`, ready for `embed`.
  //
  // **Let's simplify `RecursiveStep` and `RecursiveEmbed` for common cases.**
  //
  // For `Tree<A>`, `RecursiveStep` is `{ value: A, left: Tree<A>, right: Tree<A> }`.
  // For `Tree<A>`, `RecursiveEmbed` is `{ value: A, left_result: B, right_result: B }`.
  //
  // The `apomorphism` function will recursively call itself on `left` and `right` subtrees,
  // then pass the results and `value` to `embed`.

  // **Example 1: Sum of Tree Elements (Final Approach)**
  //
  // `T = Tree<number>`, `B = number`
  //
  // `project: (tree: Tree<number>) => number | { value: number, left: Tree<number>, right: Tree<number> }`
  //   - `EmptyTree` -> `0`
  //   - `Node(v, l, r)` -> `{ value: v, left: l, right: r }`
  //
  // `embed: (components: { value: number, left_result: number, right_result: number }) => number`
  //   - `{ value, left_result, right_result }` -> `value + left_result + right_result`
  //
  // **The `apomorphism` function will look like this:**
  //
  // `apomorphism<T, B, RecInfo>(`
  //   `project: (t: T) => B | RecInfo<T>,`
  //   `embed: (rec_results: RecInfo<B>) => B, // RecInfo<B> means RecInfo where sub-trees are replaced by results B`
  //   `value: T`
  // `): B { ... }`
  //
  // This requires `RecInfo<B>` to be structured such that the recursive parts are replaced by `B`.
  //
  // **Let's be explicit about the structure for `Tree<A>`:**
  //
  // `RecInfo<T> = { value: A, left: T, right: T }`
  // `RecInfo<B> = { value: A, left_result: B, right_result: B }` (where A is the element type of the tree)
  //
  // This means the `project` and `embed` functions must be tailored to the specific structure.
  // The challenge is to make `apomorphism` generic.

  // **Let's define the `Apomorphism` as a record containing these two functions, and then implement `apomorphism`.**
  //
  // `interface Apomorphism<T, B, RecInfoStructure>`: This is getting too complicated with type parameters.

  // **Simpler problem definition:**
  //
  // Implement an `apomorphism` function that takes:
  // 1. A `project` function: `(t: T) => B | RecursiveDecomposition<T>`
  // 2. An `embed` function: `(recursiveResults: RecursiveComposition<B>) => B`
  //
  // Where `RecursiveDecomposition<T>` and `RecursiveComposition<B>` are helper types that represent the "shape" of the recursive step.
  //
  // **For `Tree<number>` summing:**
  //
  // `RecursiveDecomposition<Tree<number>>` would be `{ value: number, left: Tree<number>, right: Tree<number> }`.
  // `RecursiveComposition<number>` would be `{ value: number, left_result: number, right_result: number }`.
  //
  // **The `apomorphism` function will require a way to map `RecursiveDecomposition<T>` to `RecursiveComposition<B>` by recursively calling itself.**

  // Let's provide concrete types for the examples.

  // **Example 1: Sum of Tree Elements**
  //
  // Input: `exampleTree: Tree<number>`
  // Output: `17`
  // Explanation: 5 + 3 + 1 + 8 = 17.

  // **Example 2: Calculating the Height of a Tree**
  //
  // Input: `exampleTree: Tree<number>`
  // Output: `3`
  // Explanation: The longest path from the root to a leaf has 3 edges (or 3 nodes depending on definition). Assuming nodes: 5 -> 3 -> 1 is path of length 3.
  //
  // `project: (tree: Tree<number>) => number | { value: number, left: Tree<number>, right: Tree<number> }`
  //   - `EmptyTree` -> `0` (height of empty tree is 0)
  //   - `Node(v, l, r)` -> `{ value: v, left: l, right: r }`
  //
  // `embed: (components: { value: number, left_result: number, right_result: number }) => number`
  //   - `{ value, left_result, right_result }` -> `1 + Math.max(left_result, right_result)` (height is 1 + max height of children)
  //
  // This means the `apomorphism` function itself needs to be generic over the recursive structure's decomposition.

  // **Let's define the `Apomorphism` contract clearly.**
  //
  // An `Apomorphism<T, B>` is defined by:
  //
  // 1. `project`: A function that deconstructs `T`. It returns either a base result `B`
  //    or a structure containing the "fields" of `T` and the recursive sub-structures.
  //    The type of this structure must be consistently representable. Let's call this `F<T>`.
  //    So, `project: (t: T) => B | F<T>`.
  //
  // 2. `embed`: A function that reconstructs the result `B` from a structure that
  //    contains the results of the recursive calls and the non-recursive parts of `T`.
  //    This structure will be `F<B>`.
  //    So, `embed: (f_of_results: F<B>) => B`.
  //
  // The `apomorphism` function will take `project` and `embed` and a value `T`, and return `B`.

  // **Concrete types for `F<T>`:**
  //
  // For `Tree<A>`:
  // `F<T> = { value: A, left: T, right: T }`
  //
  // For `List<A>`:
  // `F<T> = { head: A, tail: T }` (or `null` for empty list)

  // This requires us to know the structure of `F`. The challenge is to make `apomorphism` generic.
  //
  // **Let's assume the user provides the type for `F` that describes the recursive structure.**
  //
  // `apomorphism<T, B, F_T>(`
  //   `project: (t: T) => B | F_T<T>,`
  //   `embed: (f_of_results: F_T<B>) => B,`
  //   `value: T`
  // `): B`
  //
  // This signature seems most practical. `F_T` is a type constructor that takes `T` and returns the intermediate structure.

  // **Example 1: Sum of Tree Elements**
  //
  // `T = Tree<number>`
  // `B = number`
  // `F_T = <U>(inner: U) => { value: number, left: U, right: U }`  (This is not quite right, `F_T` needs to map to the structure containing `U`.)
  //
  // **Correcting the `F_T` concept:**
  // `F_T` should be a type that describes the recursive structure, parameterized by the "recursive element".
  //
  // For `Tree<A>`:
  // `RecursiveShape<A, Inner>` = `{ value: A, left: Inner, right: Inner }`
  //
  // So, `project` takes `Tree<A>` and returns `B | RecursiveShape<A, Tree<A>>`.
  // And `embed` takes `RecursiveShape<A, B>` and returns `B`.

  // This is the core of the problem: defining and using this `RecursiveShape`.

  // **Let's make `Apomorphism` a record of `project` and `embed`, and `apomorphism` a function that uses it.**

  // `interface Apomorphism<T, B, RecShapeT, RecShapeB>` where `RecShapeT` is `F<T>` and `RecShapeB` is `F<B>`?
  // This is still complex.

  // **Final Approach for the Challenge:**
  //
  // The challenge asks to implement apomorphisms in types. This means defining the structure of `Apomorphism` and the `apomorphism` function.
  //
  // We will define `Apomorphism` for a recursive type `T`, and a result type `B`.
  // The key is how to represent the recursive decomposition and composition.
  //
  // **`Apomorphism<T, B>` Object:**
  //   - `project`: `(t: T) => B | Projection<T>`
  //     `Projection<T>`: A structure describing the recursive breakdown. For a tree, it would be `{ value: number, left: Tree<number>, right: Tree<number> }`.
  //   - `embed`: `(composition: Composition<B>) => B`
  //     `Composition<B>`: A structure describing how to compose results from recursive calls. For a tree summing, it's `{ value: number, left_result: number, right_result: number }`.
  //
  // The `apomorphism` function will orchestrate these.

  // **Let's provide the definitions for `Tree` and `List` first.**

  // `interface EmptyTree<A> { type: 'empty'; }`
  // `interface Node<A> { type: 'node'; value: A; left: Tree<A>; right: Tree<A>; }`
  // `type Tree<A> = EmptyTree<A> | Node<A>;`

  // `interface EmptyList {}`
  // `interface Cons<A> { type: 'cons'; head: A; tail: List<A>; }`
  // `type List<A> = EmptyList | Cons<A>;`

  // **Now, define the `Apomorphism` type and the `apomorphism` function.**

  // **Example 1: Sum of Tree Elements**
  // `T = Tree<number>`, `B = number`
  //
  // `Projection<Tree<number>> = { value: number, left: Tree<number>, right: Tree<number> }`
  // `Composition<number> = { value: number, left_result: number, right_result: number }`
  //
  // `project: (tree: Tree<number>) => number | Projection<Tree<number>>`
  // `embed: (comp: Composition<number>) => number`

  // **The `apomorphism` function will need to handle the mapping from `Projection<T>` to `Composition<B>` by recursive calls.**
  // This is the critical part that needs to be generic.
  //
  // The generic `apomorphism` function needs to know how to extract recursive parts from `Projection<T>` and how to build `Composition<B>` from them.

  // **Final thought:** The problem is to implement the generic `apomorphism` *pattern*. This means defining the `Apomorphism` type that describes how to do this for *any* recursive structure (given some constraints), and then implementing the `apomorphism` function that uses this `Apomorphism` definition.

  // Let's propose a `Apomorphism` type that is parameterized by the recursive structure's decomposition.

  // `interface Apomorphism<T, B, RecStruct<S>>`:
  //   - `project: (t: T) => B | RecStruct<T>`
  //   - `embed: (rec_results: RecStruct<B>) => B`
  //
  // `apomorphism<T, B, RecStruct<S>>(apo: Apomorphism<T, B, RecStruct>, value: T): B`
  //
  // This looks like the correct path. The user will need to provide the `RecStruct` type.

  // For `Tree<A>`: `RecStruct<Inner> = { value: A, left: Inner, right: Inner }`
  // For `List<A>`: `RecStruct<Inner> = { head: A, tail: Inner }` (or similar for empty list)

  // This implies the `Apomorphism` itself needs to be constructed for each specific recursive structure.

  // Let's provide the definition for `Apomorphism` and `apomorphism` and then show how to instantiate it for the examples.

  // **The `apomorphism` function will need a mechanism to recursively process the `RecStruct<T>` and turn it into `RecStruct<B>`.**
  //
  // This mechanism is essentially a recursive call within the `apomorphism` function, parameterized by the structure of `RecStruct`.
  //
  // This is the most complex part: making the `apomorphism` function generic over the structure of `RecStruct`.

  // **Let's provide a `foldMap` equivalent for apomorphism.**
  //
  // `apomorphism<T, B, RecStruct<S>, ResultMap>`
  //
  // This is still too abstract.
  //
  // **Let's simplify the problem by focusing on the core `cata` and `ana` functions.**
  //
  // **Problem Description:**
  //
  // Implement a generic `apomorphism` function that works with recursive data structures.
  // This function should take:
  //
  // 1. `cata`: A function that deconstructs the recursive structure. It returns either a base result (`B`) or a description of the recursive step, containing the original components and sub-structures.
  // 2. `ana`: A function that reconstructs the recursive structure. It takes a "seed" (which includes results from recursive computations) and produces either a base result (`B`) or the next level of the recursive structure.
  //
  // The `apomorphism` function itself will manage the recursive calls and the transformation between the deconstructed and reconstructed states.

  // **Consider a `foldMap` like signature:**
  //
  // `apomorphism<T, B, F>(cata: (t: T) => B | F<T>, ana: (seed: F<B>) => B, value: T)`
  //
  // This requires `F` to be a type that can represent both recursive structure (`F<T>`) and composed results (`F<B>`).

  // **Let's stick to the `project`/`embed` model and define it clearly for the challenge.**

  // `project: (t: T) => B | { value: A, recursive_parts: T[] }`  (Simplified for a generic structure)
  // `embed: (value: A, recursive_results: B[]) => B`

  // This is a bit too simplified.

  // **Final decision on the approach:**
  //
  // Define an `Apomorphism` type that specifies how to `project` a recursive structure into a base case or a recursive step, and how to `embed` results into a final value.
  // Then, implement a generic `apomorphism` function that uses this `Apomorphism` definition.
  //
  // The `project` function will return either `B` or an object that *describes* the recursive structure, including the original non-recursive parts and the recursive sub-structures.
  // The `embed` function will take an object that *describes* how to compose results, including the original non-recursive parts and the *results* of recursive calls.
  //
  // The `apomorphism` function needs to handle the recursion and the transformation between these intermediate structures.

  // **Let's start with the `Apomorphism` type definition.**
};

// --- Provided Data Structures (for examples) ---

interface EmptyTree<A> {
  type: 'empty';
}

interface Node<A> {
  type: 'node';
  value: A;
  left: Tree<A>;
  right: Tree<A>;
}

type Tree<A> = EmptyTree<A> | Node<A>;

interface EmptyList {}
interface Cons<A> {
  type: 'cons';
  head: A;
  tail: List<A>;
}
type List<A> = EmptyList | Cons<A>;

// --- Apomorphism Implementation ---

/**
 * Represents the structure of a recursive decomposition.
 * For a data structure like Tree<A>, this would be a Node containing its value
 * and its left and right children, which are also Tree<A>.
 * For a List<A>, this would be a Cons cell containing its head and tail List<A>.
 *
 * The 'RecursivePart' type parameter represents the "recursive element" of the structure.
 * For a Tree<A>, RecursivePart<Tree<A>> would be { value: A, left: Tree<A>, right: Tree<A> }.
 * For a List<A>, RecursivePart<List<A>> would be { head: A, tail: List<A> }.
 */
type RecursiveDecomposition<T, RecursivePart<S>> = RecursivePart<T>;

/**
 * Represents the structure for composing results from recursive calls.
 * This mirrors the RecursiveDecomposition but with recursive results (type B)
 * instead of recursive sub-structures (type T).
 *
 * For a Tree<A> and result B, this would be { value: A, left_result: B, right_result: B }.
 */
type RecursiveComposition<B, RecursivePart<S>> = RecursivePart<B>;

/**
 * Defines the contract for an apomorphism operation on a recursive type T,
 * yielding a result of type B.
 *
 * @template T The recursive data structure type.
 * @template B The final result type.
 * @template RecPart<S> A type constructor that describes the non-recursive parts
 *   and the recursive placeholders for the structure T. For example, for a
 *   binary tree with elements of type A, it might be:
 *   <S>(value: A, left: S, right: S) => { value: A; left: S; right: S }
 *   When used with T, it becomes RecursivePart<T> (e.g., { value: A; left: T; right: T }).
 *   When used with B, it becomes RecursivePart<B> (e.g., { value: A; left_result: B; right_result: B }).
 */
interface Apomorphism<T, B, RecPart<S>> {
  /**
   * Deconstructs a value of type T.
   * - If T is a base case, it returns the final result of type B.
   * - If T is a recursive case, it returns a structure describing the recursive
   *   decomposition (RecPart<T>), containing the original non-recursive parts
   *   and the recursive sub-structures.
   */
  project: (t: T) => B | RecPart<T>;

  /**
   * Reconstructs the final result of type B from the results of recursive calls.
   * It takes a structure (RecPart<B>) that contains the original non-recursive parts
   * and the computed results (B) for the recursive sub-structures.
   */
  embed: (recResults: RecPart<B>) => B;
}

/**
 * The generic apomorphism function. It applies an apomorphic operation to a
 * recursive data structure.
 *
 * @template T The recursive data structure type.
 * @template B The final result type.
 * @template RecPart<S> The type constructor describing the recursive structure's shape.
 * @param apo An Apomorphism definition for type T yielding type B.
 * @param value The input recursive data structure of type T.
 * @returns The final result of type B.
 */
function apomorphism<T, B, RecPart<S>>(
  apo: Apomorphism<T, B, RecPart>,
  value: T
): B {
  const projected = apo.project(value);

  // Check if it's a base case (returns B directly)
  // This check is a bit loose and relies on structural typing. A more robust check
  // might involve a discriminant property if T has one. For this example, we assume
  // that if it's not the type returned by RecPart<T>, it's B.
  // A better approach would be to pass a "type guard" for B.
  // For simplicity in this challenge, we'll assume that if it's not an object that
  // structurally matches RecPart<T>, it's B.
  // In a real scenario, you might check for specific properties of RecPart<T>.

  // This check needs to be more robust. If `B` can be an object, this will fail.
  // For this challenge, we'll rely on the clear distinction in the examples.
  // A common pattern is `if (typeof projected === 'number' && !isNaN(projected))` for number results.
  // For more general types, a discriminator or a type guard function is needed.
  // Let's assume for the purpose of this challenge that `project` will return `B`
  // directly for base cases and an object for recursive steps.

  if (typeof projected === 'object' && projected !== null) {
    // Attempt to cast to RecPart<T> and process recursively
    // This is a common pattern but requires careful type management.
    // We assume `projected` has the shape of `RecPart<T>` if it's an object.
    const recursiveStep = projected as RecPart<T>;

    // This is the core of the apomorphism: recursively call `apomorphism` on sub-structures
    // and then transform the results into the format expected by `embed`.
    // This requires knowing how to extract the recursive sub-structures from `RecPart<T>`
    // and how to build `RecPart<B>` from them.

    // For generic `RecPart`, this mapping needs to be explicitly defined or inferred.
    // For this challenge, we'll provide concrete implementations for Tree and List,
    // which implicitly define this mapping.

    // This is where the generic nature of `apomorphism` becomes tricky.
    // We need a way to iterate/map over the `RecPart<T>` to get `RecPart<B>`.

    // **Let's define a helper for the recursive processing within `apomorphism`.**
    // This helper will take a `RecPart<T>` and transform it into a `RecPart<B>`.
    // This transformation *is* the recursive step of the apomorphism.

    const mapRecursiveStep = <TRef, BRef>(
      recStepT: RecPart<TRef>,
      recursiveProcessor: (value: TRef) => BRef
    ): RecPart<BRef> => {
      // This part is highly dependent on the structure of RecPart.
      // For this to be generic, RecPart would need methods or we'd need
      // a way to describe its structure (e.g., using lenses or traversals).

      // For the purpose of this challenge, we will implement this mapping
      // implicitly within the example instantiations of `apomorphism`.
      // The `apomorphism` function itself will handle the recursion.

      // The logic here is: `RecPart<T>` contains sub-values of type `T`.
      // We need to call `apomorphism` on each of these `T` values.
      // The result of `apomorphism(..., sub_t)` is `B`.
      // We then collect these `B` results into a `RecPart<B>`.

      // This requires introspection or explicit handling of `RecPart`'s fields.
      // Since we don't have a generic way to do this for arbitrary `RecPart`,
      // the `apomorphism` implementation below will be slightly tailored to
      // how `RecPart` is defined for the examples.

      // Let's assume the `RecPart` interface can be traversed.
      // This is where type-level programming or runtime reflection would be useful.

      // **Simplified Strategy:**
      // The `apomorphism` function will directly handle the recursion.
      // The `project` function dictates the structure of `RecPart<T>`.
      // The `embed` function dictates how to form `B` from `RecPart<B>`.
      // The `apomorphism` function itself will map `RecPart<T>` to `RecPart<B>` by
      // recursively calling itself on the `T` parts within `RecPart<T>`.

      // This means `apomorphism` needs to "know" how to extract the recursive `T` values
      // from `RecPart<T>` and how to build `RecPart<B>` from the `B` results.
      // This is the most challenging aspect of making `apomorphism` truly generic.

      // For the examples below, the `apomorphism` implementation will be specific to the
      // structure of `RecPart` for Trees and Lists.

      // If `RecPart` is an object, and we can assume its fields are either `B` or `T`,
      // we can try to map. This is unsafe without a clear type definition for `RecPart`.

      // **Let's refine the `apomorphism` implementation to show the recursive calls explicitly.**
      // The structure of `RecPart` will be defined for each example.

      // This is a placeholder for the complex mapping logic.
      // The actual recursive calls will be structured within the example instantiations.
      throw new Error("Recursive processing logic needs specific implementation for RecPart structure.");
    };

    // The actual logic will be structured in the specific examples below.
    // For a generic `apomorphism` function, you'd need a way to describe `RecPart`'s structure.
    // For instance, if `RecPart` is a record, you'd iterate over its values.
    // If a value is of type `T`, recursively call `apomorphism`.
    // If a value is of type `B`, keep it.
    // Then, use `apo.embed` with the resulting structure.

    // This is a conceptual implementation. The actual mapping from `RecPart<T>` to `RecPart<B>`
    // is the difficult part for a truly generic `apomorphism`.

    // The following structure assumes a way to map `RecPart<T>` to `RecPart<B>`.
    // We will simulate this mapping by explicit recursive calls in the examples.
    // This demonstrates the *concept* of apomorphism.

    // Conceptual reconstruction:
    // const mappedRecPart = mapRecursiveStep(recursiveStep, (subT) => apomorphism(apo, subT));
    // return apo.embed(mappedRecPart);

    // Given the complexity of a truly generic `mapRecursiveStep` without advanced type system features or libraries,
    // we will make the `apomorphism` function slightly less abstract and show how it works for specific `RecPart` structures.

    // The `apomorphism` function will be implemented directly in the example usage sections.
    // This makes the example clearer by showing the recursion in context.

    // Returning a dummy value here as the actual implementation is complex and example-specific.
    // The examples below will demonstrate the intended usage and logic.
    throw new Error("Generic apomorphism function implementation requires more specific context on RecPart structure.");
  } else {
    // If `projected` is not an object, it must be the base result `B`.
    return projected as B;
  }
}


## Examples

To illustrate apomorphisms, we will implement them for `Tree<A>` and `List<A>`.

### Example 1: Summing elements in a Tree

This example demonstrates how to use apomorphism to calculate the sum of all numeric elements in a binary tree.

**Data Structure:** `Tree<A>`
**Result Type:** `number`

**1. Define the `RecPart` types for `Tree<A>`:**

For `Tree<A>`, the recursive decomposition (`RecPart<T>`) is a node with its value and children:
`type TreeRecPart<T> = { value: A; left: T; right: T; }`

And the recursive composition (`RecPart<B>`) for results is:
`type TreeComp<B> = { value: A; left_result: B; right_result: B; }`

**2. Define the `Apomorphism` for summing a `Tree<number>`:**

```typescript
// Define RecPart for Tree<number>
type TreeNumRecPart<S> = { value: number; left: S; right: S };

const sumTreeApomorphism: Apomorphism<Tree<number>, number, TreeNumRecPart> = {
  project: (tree) => {
    if (tree.type === 'empty') {
      return 0; // Base case: sum of an empty tree is 0
    } else {
      // Recursive case: return the node's value and sub-trees
      return { value: tree.value, left: tree.left, right: tree.right };
    }
  },
  embed: (composition) => {
    // Combine the current node's value with the results from its children
    return composition.value + composition.left_result + composition.right_result;
  },
};

// Implement the apomorphism function for this specific case
function apomorphismSumTree(
  value: Tree<number>
): number {
  const projected = sumTreeApomorphism.project(value);

  if (typeof projected === 'number') {
    return projected; // Base case
  } else {
    // Recursive case: project is { value, left, right }
    const { value: nodeValue, left, right } = projected;

    // Recursively process left and right subtrees
    const leftResult = apomorphismSumTree(left);
    const rightResult = apomorphismSumTree(right);

    // Use embed to combine the current node's value with the results
    return sumTreeApomorphism.embed({ value: nodeValue, left_result: leftResult, right_result: rightResult });
  }
}

Input:

const exampleTree: Tree<number> = {
  type: 'node',
  value: 5,
  left: {
    type: 'node',
    value: 3,
    left: {
      type: 'node',
      value: 1,
      left: { type: 'empty' },
      right: { type: 'empty' },
    },
    right: { type: 'empty' },
  },
  right: {
    type: 'node',
    value: 8,
    left: { type: 'empty' },
    right: { type: 'empty' },
  },
};

Output:

17

Explanation: The apomorphismSumTree function deconstructs the tree.

• For EmptyTree, it returns 0.
• For Node(5, leftTree, rightTree), it recursively calls itself on leftTree and rightTree, obtaining leftResult and rightResult. Then, it uses sumTreeApomorphism.embed with { value: 5, left_result: leftResult, right_result: rightResult } to compute 5 + leftResult + rightResult. This process unwinds to sum all elements.

Example 2: Calculating the Height of a Tree

This example demonstrates calculating the height of a binary tree.

Data Structure: Tree<A> Result Type: number

1. Define RecPart types for Tree<A> (same as Example 1).

2. Define the Apomorphism for calculating the height of a Tree<number>:

// Use the same TreeNumRecPart as in Example 1
// type TreeNumRecPart<S> = { value: number; left: S; right: S };

const heightTreeApomorphism: Apomorphism<Tree<number>, number, TreeNumRecPart> = {
  project: (tree) => {
    if (tree.type === 'empty') {
      return 0; // Base case: height of an empty tree is 0
    } else {
      // Recursive case: return the node's value and sub-trees.
      // The value itself isn't used for height, but the structure is needed.
      return { value: tree.value, left: tree.left, right: tree.right };
    }
  },
  embed: (composition) => {
    // The height of a node is 1 + the maximum height of its children.
    return 1 + Math.max(composition.left_result, composition.right_result);
  },
};

// Implement the apomorphism function for this specific case
function apomorphismHeightTree(
  value: Tree<number>
): number {
  const projected = heightTreeApomorphism.project(value);

  if (typeof projected === 'number') {
    return projected; // Base case
  } else {
    // Recursive case: project is { value, left, right }
    const { value: nodeValue, left, right } = projected; // nodeValue is ignored here

    // Recursively calculate the height of left and right subtrees
    const leftHeight = apomorphismHeightTree(left);
    const rightHeight = apomorphismHeightTree(right);

    // Use embed to combine the results: 1 + max height of children
    return heightTreeApomorphism.embed({ value: nodeValue, left_result: leftHeight, right_result: rightHeight });
  }
}

Input:

// Using the same exampleTree from Example 1

Output:

3

Explanation: The apomorphismHeightTree function deconstructs the tree.

• For EmptyTree, it returns 0.
• For Node(value, leftTree, rightTree), it recursively calls itself on leftTree and rightTree to get their heights (leftHeight, rightHeight). Then, it uses heightTreeApomorphism.embed with { value: ..., left_result: leftHeight, right_result: rightHeight } to compute 1 + Math.max(leftHeight, rightHeight).

Example 3: Reversing a List

This example demonstrates using apomorphism to reverse a list. This is a good example of how apomorphism can also be used for construction or reconstruction.

Data Structure: List<A> Result Type: List<A>

1. Define the RecPart types for List<A>:

For List<A>, the recursive decomposition (RecPart<T>) is a cons cell: type ListRecPart<T> = { head: A; tail: T; } Note: An EmptyList is handled as a base case.

And the recursive composition (RecPart<B>) for results (which are also lists): type ListComp<B> = { head: A; tail_result: B; } (Here, tail_result is the reversed tail list).

2. Define the Apomorphism for reversing a List<number>:

// Define RecPart for List<number>
type ListNumRecPart<S> = { head: number; tail: S };

const reverseListApomorphism: Apomorphism<List<number>, List<number>, ListNumRecPart> = {
  project: (list) => {
    if (list && typeof list === 'object' && 'type' in list && list.type === 'cons') {
      // Recursive case: it's a Cons cell
      return { head: list.head, tail: list.tail };
    } else {
      // Base case: it's an EmptyList
      return {}; // Return an empty object for the base case, representing an empty list
    }
  },
  embed: (composition) => {
    // This is the crucial part for reversal.
    // `composition` is { head: number, tail_result: List<number> }
    // We need to prepend the 'head' to the 'tail_result' (which is the reversed tail).
    // This is effectively building the reversed list from the end.
    //
    // However, the standard apomorphism `embed` typically builds the structure `T`.
    // For reversal, we are transforming `List<number>` into `List<number>`.
    // The 'head' needs to be placed at the *end* of the reversed tail.
    // This suggests a pattern where `embed` takes the original head and the reversed tail.
    //
    // For reversal, the standard `embed` would reconstruct the original structure.
    // The `apomorphism` function's recursive calls need to handle the transformation.

    // Let's rethink `embed` for reversal.
    // When `project` returns `{ head: h, tail: t }`, `apomorphism` calls itself on `t` to get `reversed_tail`.
    // Then `embed` needs to combine `h` and `reversed_tail`.
    // The result of `embed` should be `List<number>`.
    //
    // The standard way to reverse a list recursively without extra space often uses an accumulator.
    // Apomorphism can achieve this by using the `embed` step to append to the result of the reversed tail.
    //
    // `embed({ head: current_head, tail_result: reversed_tail })`
    // We need to append `current_head` to `reversed_tail`.
    // This means `embed` should return `List<number>`.

    // A common recursive list reversal pattern:
    // reverse(list):
    //   if list is empty: return empty list
    //   else:
    //     head = list.head
    //     tail = list.tail
    //     reversed_tail = reverse(tail)
    //     return append(reversed_tail, head)
    //
    // In our apomorphism:
    // `project(list)` gives `{ head, tail }` or `{}`.
    // `apomorphism(list)` will recursively call on `tail` to get `reversed_tail`.
    // Then `embed` will take `{ head, tail_result: reversed_tail }` and return `append(reversed_tail, head)`.

    // This requires an `append` function for lists.
    const append = (list1: List<number>, element: number): List<number> => {
      if (!list1 || (typeof list1 === 'object' && Object.keys(list1).length === 0)) {
        // If list1 is EmptyList ({})
        return { type: 'cons', head: element, tail: {} };
      } else {
        // If list1 is Cons
        const { head, tail } = list1 as Cons<number>;
        return {
          type: 'cons',
          head: head,
          tail: append(tail, element),
        };
      }
    };

    return append(composition.tail_result, composition.head);
  },
};

// Implement the apomorphism function for this specific case
function apomorphismReverseList(
  list: List<number>
): List<number> {
  const projected = reverseListApomorphism.project(list);

  if (Object.keys(projected).length === 0 && typeof projected === 'object') {
    // Base case: EmptyList
    return {}; // Return an empty list representation
  } else {
    // Recursive case: project is { head, tail }
    const { head, tail } = projected as ListNumRecPart<List<number>>;

    // Recursively reverse the tail of the list
    const reversedTail = apomorphismReverseList(tail);

    // Use embed to prepend the current head to the reversed tail.
    // The 'embed' function's responsibility is to construct the result.
    // Here, 'composition' is { head, tail_result: reversedTail }
    return reverseListApomorphism.embed({ head: head, tail_result: reversedTail });
  }
}

Input:

const exampleList: List<number> = {
  type: 'cons',
  head: 1,
  tail: {
    type: 'cons',
    head: 2,
    tail: {
      type: 'cons',
      head: 3,
      tail: {}, // EmptyList
    },
  },
};

Output:

{
  type: 'cons',
  head: 3,
  tail: {
    type: 'cons',
    head: 2,
    tail: {
      type: 'cons',
      head: 1,
      tail: {}
    }
  }
}

Explanation: The apomorphismReverseList function deconstructs the list.

• For an EmptyList ({}), it returns an EmptyList.
• For a Cons(h, t), it recursively calls itself on t to get reversedTail. Then, it uses reverseListApomorphism.embed({ head: h, tail_result: reversedTail }). The embed function's logic append(reversedTail, h) takes the reversed tail and appends the current head to its end, effectively building the reversed list.

Constraints

• The solution must be written in TypeScript.
• All type definitions and functions should be clear and well-commented.
• The implementation should be as generic as possible, demonstrating the core concepts of apomorphisms.
• For the examples, assume standard JavaScript object structures and type checks are sufficient for distinguishing base cases from recursive cases where applicable.

Notes

• Apomorphisms generalize catamorphisms (folds) and anamorphisms (unfolds). While catamorphisms deconstruct and anamorphisms construct, apomorphisms can do both.
• The project function handles the deconstruction (similar to cata), and embed handles the reconstruction (similar to ana).
• The key challenge in a truly generic implementation is managing the RecPart structure, which describes the shape of the recursive decomposition and composition. In practice, libraries often use type-level programming or combinators to handle this generically. For this challenge, you are expected to show the pattern by providing concrete Apomorphism instances for specific structures.
• The examples provided use simplified TypeScript interfaces for recursive data structures. You may need to adapt your type definitions based on how you represent these structures.
• Pay close attention to the base cases for each data structure and operation.