Problem A: Let T be a balanced 5-ary tree with 81 nodes.
(a). How many internal nodes does T have?
(b). How many edges does the tree T have?
(c). What is the height of T?
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To answer these questions, we first need to understand the properties of a balanced 5-ary tree.
In a balanced 5-ary tree:
- Each node can have 0 to 5 children.
- The tree is perfectly balanced, meaning the height of all the subtrees differ by at most 1 level.
Now let's solve each part of the problem step by step:
(a) How many internal nodes does T have?
- An internal node is any node that has at least one child. In other words, internal nodes are all nodes except the leaf nodes.
- To determine the number of internal nodes in a balanced 5-ary tree, we can use the formula:
(Number of nodes - Number of leaf nodes) / Number of children per node
In this case, the number of children per node is 5 in a 5-ary tree.
- T has 81 nodes. To find the number of leaf nodes, we need to determine the height of the tree.
- Let's move on to part (c) to find the height of T.
(c) What is the height of T?
- The height of a balanced 5-ary tree can be found using the formula:
log5(number of nodes)
In this case, we have 81 nodes.
So, the height of T is log5(81).
Now let's calculate the height of T:
- log5(81) approximately equals 2.622.
- Since the height cannot be a decimal value, we round it up to the nearest integer.
- Hence, the height of T is 3.
Now that we know the height of T, we can calculate the number of leaf nodes:
- Leaf nodes are the nodes present at the last level of the tree.
- In a balanced 5-ary tree, the maximum number of leaf nodes at any level is the number of children per node raised to the power of the height of the tree.
- For T, the height is 3, and the number of children per node is 5.
- Hence, the maximum number of leaf nodes at level 3 is 5^3 = 125.
- Since we have only 81 nodes in T, the actual number of leaf nodes is 81.
Now, we can calculate the number of internal nodes:
- (Number of nodes - Number of leaf nodes) / Number of children per node
- (81 - 81) / 5 = 0
- Therefore, T has 0 internal nodes.
(b) How many edges does the tree T have?
- The number of edges in a tree is equal to the number of nodes minus 1.
- So, the number of edges in T is 81 - 1 = 80.
To summarize:
(a) T has 0 internal nodes.
(b) T has 80 edges.
(c) The height of T is 3.