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other options because the right subtree must be a singleton. Therefore, there are five different binary trees of size 5 whose left subtree has size 3. Next, count all the binary trees of size 5 whose left subtree has size 2. There are only two different possibilities for that left subtree. But for each of those two choices, we have the same two different possibilities for the right subtree because it also must have size 2. Therefore, there are 2 2 = 4 different binary trees of size 5 whose left subtree has size 2. By similar reasoning, we find that there are five different binary trees of size 5 whose left subtree has size 1, and there are 14 different binary trees of size 5 whose left subtree has size 0. Therefore, the total number of different binary trees of size 5 is 14 + 5 + 4 + 5 + 14 = 42.

The instantaneous power ow into the port is given by p t v t i t Vm Im cos !t cos !t 1 Vm Im cos 2!t cos 2 By (1.20), P0 1 2 2

FULL BINARY TREES A binary tree is said to be full if all its leaves are at the same level and every interior node has two children. EXAMPLE 11.5 The Full Binary Tree of Height 3

2

Euclid discovered an algorithm that systematically and quickly reduces the size of the problem by replacing the original pair of numbers by smaller pairs until one of the pair becomes zero, at which point the GCD is the other number of the pair (the GCD of any number and 0 is that number) Here is Euclid s algorithm for finding the GCD of any two numbers A and B Repeat: If B is zero, the GCD is A.

The tree shown in Figure 11.6 is the full binary tree of height 3. Note that it has 15 nodes: 7 interior nodes and 8 leaves.

After the integration is performed and its limits evaluated, the result is P0 V m Im V Im cos pm p cos VI cos 2 2 2

Theorem 111 The full binary tree of height h has l = 2h leaves and m = 2h 1 internal nodes Proof: The full binary tree of height h = 0 is a single leaf node, so it has n = 1 node, which is a leaf Therefore, since 2h 1 = 20 1 = 1 1 = 0, and 2h = 20 = 1, the formulas are correct for the case where h = 0 More generally, let h > 0 and assume (the inductive hypothesis) that the formulas are true for all full binary trees of height less than h Then consider a full binary tree of height h Each of its subtrees has height h 1, so we apply the formulas to them: lL = lR = 2h 1 and mL = mR = 2h 1 1.

CHAP. 1]

Otherwise: find the remainder R when dividing A by B replace the value of A with the value of B replace the value of B with the value of R For example, to find the GCD of 372 and 84, which we will show as: GCD(372, 84) Find GCD(84, 36) because 372/84 > remainder 36 Find GCD(36, 12) because 84/36 > remainder 12 Find GCD(12, 0) because 36/12 > remainder 0; Solved! GCD = 12 More formally, an algorithm is a sequence of computations that operates on some set of inputs and produces a result in a finite period of time In the example of the algorithm for designing stairs, the inputs are the total rise and total run The result is the best specification for the number of steps, and for the rise and run of each step.

1.23 Prove that the capacitor element of Fig. 1-1(c) is a linear element by showing that it satis es the converse of the superposition theorem. (Hint: See Problem 1.1.) Use the superposition theorem to nd the current i in Fig. 1-14 if R1 5 ; R2 10 ; Vs 10 cos 2t V, and Is 3 cos 3t =4 A. Ans: i 0:667 cos 2t cos 3t =4 A In Fig. 1-23, (a) nd the Thevenin equivalent voltage and impedance for the network to the left of terminals a; b, and (b) use the Thevenin equivalent circuit to determine the current IL . Ans: a VTh V1 I2 R2 ; ZTh R1 R2 ; b IL V1 I2 R2 = R1 R2 RL

(These are the number of leaves in the left subtree, the number of leaves in the right subtree, the number of internal nodes in the left subtree, and the number of internal nodes in the right subtree, respectively) Then l = lL + lR = 2h 1 + 2h 1 = 2 2h 1 = 2h and m = mL + mR + 1 = (2h 1 1) + (2h 1 1) + 1 = 2 2h 1 1 = 2I 1 Therefore, by the (Second) Principle of Mathematical Induction, the formulas must be true for full binary trees of any height h 0 By simply adding the formulas for m and l, we obtain the first corollary Corollary 111 The full binary tree of height h has a total of n = 2h+1 1 nodes.

Fig. 1-23