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

The Array List Representation of a Complete Binary Tree
The array-list binary tree representation (Section 7.3.5) is especially suitable for a
complete binary tree T. We recall that in this implementation, the nodes of T are
stored in an array list A such that node v in T is the element of A with index equal
to the level number p(v) of v, defined as follows:
• If v is the root of T, then p(v) = 1.
• If v is the left child of node u, then p(v) = 2p(u).
• If v is the right child of node u, then p(v) = 2p(u) + 1.
With this implementation, the nodes of T have contiguous indices in the range
[1,n] and the last node of T is always at index n, where n is the number of nodes
of T. Figure 8.5 shows two examples illustrating this property of the last node.
Figure 8.5: Two examples showing that the last
node w of a heap with n nodes has level number n: (a)
heap T 1 with more than one node on the bottom level;
(b) heap T 2 with one node on the bottom level; (c)
array-list representation of T 1 ; (d) array-list
representation of T 2 .
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