Tree

all_schroeder_tree(num_leaves: int, num_internal_nodes: int = None) → list[Node]

Reruns all Schröder trees with given numbers of leaves and internal nodes.

Parameters:

• num_leaves (int) – Number of leaves

• num_internal_nodes (int) – Number of internal nodes

Returns:

List of the tree roots

Return type:

list[Node]

caterpillar_tree(num_leaves: int, seed: int = None) → Node

Generates a caterpillar tree. In the special case of num_leaves == 1 then we output a single node even if it is strictly speaking not a caterpillar tree.

Returns:

The root of the tree

Return type:

Node

schroeder_tree(num_leaves: int, num_internal_nodes: int | None = None, seed: int = None) → Node

Generates a random Schöder tree with suitable number of leaves and internal nodes. The algorithm was presented by Alonso, Rémy, Schott (1997a) building on Alonso, Rémy, Schott (1997b).

If the number of internal node is not give, it is selected at random such that each number of internal nodes is sampled with probability proportional to the number of Schroöder trees with the corresponding number of internal nodes.

Parameters:

• num_leaves (int) – The number of leaves of the tree

• num_internal_nodes (int | None) – The number of internal nodes of the tree

• seed (int, default: None) – Seed for numpy random number generator

Returns:

The root of the tree

Return type:

Node

Validation

This sampler is supposed to yield a uniform distribution, but does not in practice.

References

Uniform generation of a Schröder tree, L. Alonso, J.L. Rémy and R. Schott, Information Processing Letters, Volume 64, Issue 6, 1997.

A linear-time algorithm for the generation of trees, L. Alonso, J.L. Rémy and R. Schott, Algorithmica 17, 162–182, 1997

schroeder_tree_brute_force(num_leaves: int, num_internal_nodes: int | None = None, seed: int = None) → Node

Sample a Schröder tree uniformly at random by enumerating all possible Schröder tree and selecting one of them uniformly at random. This procedure is particularly inefficient but ensures uniformity.

Parameters:

• num_leaves (int) – The number of leaves of the tree

• num_internal_nodes (int | None) – The number of internal nodes of the tree

• seed (int, default: None) – Seed for numpy random number generator

Returns:

The root of the tree

Return type:

Node

Validation

This sampler yields a uniform probability distribution but enumerates all trees (and is thus slow).

schroeder_tree_lescanne(num_leaves: int, num_internal_nodes: int | None = None, seed: int = None) → Node

Samples a random Schröder tree following the algorithm provided by Lescanne (2022). The sampler is directly taken from the corresponding GitHub repository (no licence available).

If a specific number of internal nodes is given, trees are sampled at random until the required number of internal nodes is achieved (rejection sampling).

Parameters:

• num_leaves (int) – The number of leaves of the tree

• num_internal_nodes (int | None) – The number of internal nodes of the tree

• seed (int, default: None) – Seed for numpy random number generator

Returns:

The root of the tree

Return type:

Node

Validation

This sampler has unknown probability distribution but is efficient for large numbers of leaves.

References

Holonomic equations and efficient random generation of binary trees, Pierre Lesscanne, Discrete Mathematics & Theoretical Computer Science, vol. 25:2, 2024.