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A node with at least one child is called an internal node A path in a tree is a sequence of nodes (x0, x1, x2, , xm) wherein the nodes of each pair with adjacent subscripts (xi 1, xi) are adjacent nodes For example, (a, b, c, d) is a path in the tree shown above, but (a, d, b, c) is not The length of a path is the number m of its adjacent pairs It follows from the definition that trees are acyclic, that is, no path can contain the same node more than once A root path for a node x0 in a tree is a path (x0, x1, x2, , xm) where xm is the root of the tree A root path for a leaf node is called a leaf-to-root path Theorem 101 Every node in a tree has a unique root path.

If two people speak at once, we expect both to stop speaking, and then listen for the other to begin again, after a short interval Computer protocols are sometimes quite analogous to such human behavior Computer networks can be classified based on their size A small network that connects computers contained in a single room, floor, or a single building is called a local area network (LAN) LANs have become very commonplace, and you may have a LAN in your home A LAN is typically owned by an individual and connects computers that are relatively close together A wide area network (WAN) covers a large geographical distance For example a WAN may span several cities, several states, or an entire country A WAN, unlike a LAN, is typically owned and maintained by an organization and has the capacity to carry large amounts of data.

Substitution of (1) into (3) gives the current-gain ratio as hfe RB IL RC Ii RC RL RB hie ZE Using (2) in (4) and rearranging leads to the desired current-gain ratio: hfe RC RB sRE CE 1 IL RC RL RE RB hie Ai s R C R hie Ii 1 s E E B RE RB hie (b) The low-frequency current-gain ratio follows from letting s ! 0 in (5): Ai 0 lim (c) hfe RC RB IL Ii RC RL RE RB hie 6 4

For a proof, see Problem 101 on page 194 The depth of a node in a tree is the length of its root path Of course, the depth of the root in any tree is 0 We also refer to the depth of a subtree in a tree, meaning the depth of its root A level in a tree is the set of all nodes at a given depth The height of a tree is the greatest depth among all of its nodes By definition, the height of a singleton is 0, and the height of the empty tree is 1 For example, the tree A, shown in Figure 102, has height 2 Its subtree C has height 1, and its two subtrees B and D each have height 0.

The midfrequency current-gain ratio is obtained by letting s ! 1 in (5): Ai 1 lim hfe RC RB IL Ii RC RL RB hie 7

A node y is said to be an ancestor of another node x if it is on x s root path Note that the root of a tree is an ancestor of every other node in the tree A node x is said to be a descendant of another node y if y is an ancestor of x For each node y in a tree, the set consisting of y and all its descendants form the subtree rooted at y If S is a subtree of T, then we say that T is a supertree of S The path length of a tree is the sum of the lengths of all paths from its root This is the same as the weighted sum, adding each level times the number of nodes on that level.

To understand the difference between a LAN and a WAN, consider for a moment the pipes that carry water to your house Inside a typical home, water is carried in pipes that are from one-half to three-quarters of an inch in diameter The pipes are part of the house, and if one springs a leak, the owner of the house is responsible for fixing the problem The pipes buried under the street that supply the water to your house, however, are owned by the town in which you live and if they spring a leak, the town is responsible for making the repair..

(d) Inspection of (5) shows that the Laplace-domain transfer function is of the form Ai s Kb 1 s 1 2 s 1 1 R RB hie E 2 RE CE RB hie

The path length of the tree shown here is 1 3 + 2 4 + 3 8 = 35 EXAMPLE 102 Properties of a Tree.

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