The size of a binary tree is its number of internal nodes.

Binary trees of size 1: (the root is on the top and the leafs are not represented)

The number of linear extensions of a binary tree.

The depth/height of a binary tree.

The size of the left subtree of a binary tree.

The number of nodes on the left branch of a binary tree.

The number of elements smaller than a binary tree in Tamari order.

The number of left oriented leafs of a binary tree except the first one.

The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary ....

The number of occurrences of the contiguous pattern [.,[.,[.,[.,.]]]] in a bin....

The number of occurrences of the contiguous pattern [.,[.,[[.,.],.]]] in a bin....

The number of occurrences of the contiguous pattern [.,[[[.,.],.],.

The number of occurrences of the contiguous pattern [.,[.,[.,[.,[.,.]]]]] in a....

The number of occurrences of the contiguous pattern [.,[.,[.,[[.,.],.]]]] in a....

The number of occurrences of the contiguous pattern [.,[.,[[.,[.,.]],.]]] in a....

The number of occurrences of the contiguous pattern [.,[.,[[[.,.],.],.]]] in a....

The number of occurrences of the contiguous pattern [.,[[.,.],[[.,.],.]]] in a....

The number of occurrences of the contiguous pattern [.,[[[[.,.],.],.],.

The number of occurrences of the contiguous pattern [[.,.],[.,[[.,.],.]]] in a....

The sum of the sizes of the right subtrees of a binary tree.

The number of occurrences of the contiguous pattern [[.,.],[.,.

A decimal representation of a binary tree as a code word.

The number of leaf nodes in a binary tree.

The number of external nodes of a binary tree.

The number of internal nodes of a binary tree.

The number of nodes of degree 3 of a binary tree.

The number of vertices with out-degree 1 in a binary tree.

The register function (or Horton-Strahler number) of a binary tree.

The sum of the depths of the vertices (or total internal path length) of a binary....

The external path length of a binary tree.

The number of pitchforks in a binary tree.

The tree factorial of a binary tree.

The number of binary trees with the same underlying unordered tree.

The binary logarithm of the number of binary trees with the same underlying unord....

The hook number of a binary tree.

The sum of the heights of the vertices of a binary tree.

The protection number of a binary tree.

The number of maximal left branches of a binary tree.

to ordered tree: left child = left brother

to Dyck path: up step, left tree, down step, right tree

to 132-avoiding permutation

to ordered tree: right child = right brother

to 312-avoiding permutation

to Tamari-corresponding Dyck path

to Tamari-corresponding binary tree

to binary tree: up step, left tree, down step, right tree

to binary tree: left brother = left child

to binary tree: right brother = right child

binary search tree: left to right

Zeilberger's Strahler bijection

logarithmic height to pruning number

pruning number to logarithmic height