Question

Let T be a binary tree with n nodes and let f() be the level numbering function of the positions of T

f suggests a epteseniatñion of a binary tree Tty in el marabering function f suggests a f an aray-ased wucture A. with the lt of the array We show an etample of an an el ermbering funcion f sugests a represeuani sl Wr show an example of an antay baed rerjesctanisa od a A with the clement an positn pot T stoned at The lene adex fip) of the array We show a ele 12 0 1 23469 10 1 12 13 14 re &.10: Representation of a binary tree by means of an anay of an array-based representation of a binary tree is that a posi- pcan be represented by the single integer fp), and that postion-based net- ach as root, parent, left, and right can be implemented using simple anithmetic persouns oe the mamber f(p). Based on our formula for the level mumbering the ea dhid of p has index 2f(p)+ 1, the right child of p has indes 2fiP)2 and tie parent of p has index tf() 1)/2. We leave the details of a complete amay- one advantage in lsed inplementation as Exercise R-8.14 The space usage of an array based representation depends greatly os the stupe of the trec. Let n be the number of nodes of T, and et fu be the maximam value ol fp over all the nodes of T. The array A requires length NI+ fu, since denents range from Alo! to ALfu]. Note that A may lave a number of empty cels do at refer to existing positions of T. In fact, in the worst case, -1 ㈦estification of which is left as an exercise (R-812). In Section93, we will of binary trees, called "heaps for which N. Thas, in spite of the space usage, there are applications for which the array repeesentation r tee is space efficient. Siüll, for general binary trees, the exponential we a class of cve space requirement of this representation is prohibitive. eoou cannot be etticiet of an array representation is that many update operations for beciently supported. For example, removing a pode and promoting ies On time because it is not just the child that moves locations within eray but all descendants of that child