Data Structures and Algorithms in Java[1].pdf - Fulvio Frisone

same depth d as the level dof T. In a binary tree, level 0 has at most one node (the
root), level 1 has at most two nodes (the children of the root), level 2 has at most
four nodes, and so on. (See Figure 7.12.) In general, level d has at most 2 d nodes.
Figure 7.12: Maximum number of nodes in the levels
of a binary tree.
We can see that the maximum number of nodes on the levels of a binary tree grows
exponentially as we go down the tree. From this simple observation, we can derive
the following properties relating the height of a binary T with its number of nodes.
A detailed justification of these properties is left as an exercise (R-7.15).
Proposition 7.10: Let T be a nonempty binary tree, and let n, n E , n I and h
denote the number of nodes, number of external nodes, number of internal nodes,
and height of T, respectively. Then T has the following properties:
1. h+1 ≤ n ≤ 2 h+1 −1
2. 1≤n E ≤2 h
3. h≤n I ≤2 h −1
4. log(n+1)−1 ≤h≤n−1.
Also, if T is proper, then T has the following properties:
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