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Definition & Example

A binary tree is a rooted tree where each node is either internal (has two children) or is a leaf (has no children).

Equivalently, a binary tree is recursively defined to be either an empty tree (leaf) or an ordered pair of binary trees (internal node).
the 5 Binary trees of size 3  
[.,[.,[.,.]]]  [.,[[.,.],.]]  [[.,.],[.,.]]  [[.,[.,.]],.]  [[[.,.],.],.] 

The graphical representation omits the leafs and only shows the internal nodes.

There are $\operatorname{Cat}(n) = \frac{1}{n+1}\binom{2n}{n}$ binary trees with $n$ internal nodes, see OEIS:A000108.
Additional information
Feel free to add further information on the combinatorics of binary trees here!
References
Sage examples
Technical information for database usage
 A binary tree is uniquely represented as a dot (empty tree) or as a sorted list of binary trees.
 Binary trees are graded by the number of internal nodes.
 The database contains all binary trees of size at most 8.
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