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By Gilles Brassard

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1. This structure is interesting because it allows efficient searches for values in the tree. 1. Suppose the value sought is held in a node at depth p in a search tree. Design an algorithm capable of finding this node starting at the root in a time in the order of p. a It is possible to update a search tree, that is, to delete nodes or to add new values, without destroying the search tree property. However, if this is done in an unconsidered fashion, it can happen that the resulting tree becomes badly unbalanced, in the sense that the height of the tree is in the order of the number of nodes it contains.

3 Rooted Trees Let G be a directed graph. If there exists in G a vertex r such that every other vertex can be reached from r by a unique path, then G is a rooted tree and r is its root. Any rooted tree with n nodes contains exactly n - 1 edges. 3. In this example alpha is at the root of the tree. ) Extending the analogy with a family tree, we say that beta is the parent of delta and the child of alpha, that epsilon and zeta are the siblings of delta, that alpha is an ancestor of epsilon, and so on.

If each node of a rooted tree can have up to n children, we say it is an n-ary tree. In this case, the positions occupied by the children are significant. 6 are not the same: in the first case b is the elder child of a and the younger child is missing, whereas in the second case b is the younger child of a and the elder child is missing. In the important case of a binary tree, although the metaphor becomes somewhat strained, we naturally tend to talk about the left-hand child and the right-hand child.

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