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A binary tree, in the data structure convention used here, is a rooted tree in which each vertex has zero, one or two children in a rooted tree, occupying distinct left and right positions. A single left child and a single right child give different binary trees. A vertex with no children in a rooted tree is called a leaf. An empty binary tree has no root of a tree and is also allowed as a storage case. ^[ref]

For example, give the root of a tree label $4$, its left child label $2$ and its right child label $6$. Attach $1,3$ as the left and right children in a rooted tree of $2$, and $5,7$ as those of $6$. This binary tree has seven vertices, six edges, and height of a binary tree $2$: a longest downward graph path contains two edges. The first diagram below shows this shape alongside the binary tree consisting only of successive right children $1,2,\dots ,7$, whose height of a binary tree is $6$.

Left and right positions specify an ordered shape, not a numerical total order on stored labels. Swapping labels $1$ and $7$ in the first example still gives a binary tree, but destroys the binary search tree property. Nor must a binary tree be balanced: the right-child chain is just as valid. The same shape convention supports a binary heap, a Huffman code, or a binary search tree, each with its own additional rules. The useful idea is to describe two smaller subtrees at each vertex, making structural recursion natural.

A rooted tree is a tree with a distinguished vertex called the root. Its levels consist of the vertices at a fixed graph distance from the root.