General Computer Science 320201 GenCS I & II Lecture ... - Kwarc

Therefore we look at special configurations for combinatory circuits that have good depth and cost.
These will become important, when we build actual combinational circuits with given input/output
behavior.
Balanced Binary Trees
Balanced Binary Trees
Definition 395 (Binary Tree) A binary tree is a tree where all nodes have out-degree 2
or 0.
Definition 396 A binary tree G is called balanced iff the depth of all leaves differs by at
most by 1, and fully balanced, iff the depth difference is 0.
Constructing a binary tree Gbbt = 〈V, E〉 with n leaves
step 1: select some u ∈ V as root, (V1 := {u}, E1 := ∅)
step 2: select v, w ∈ V not yet in Gbbt and add them, (Vi = Vi−1 ∪ {v, w})
step 3: add two edges 〈u, v〉 and 〈u, w〉 where u is the leftmost of the shallowest nodes
with outdeg(u) = 0, (Ei := Ei−1 ∪ {〈u, v〉, 〈u, w〉})
repeat steps 2 and 3 until i = n (V = Vn, E = En)
Example 397 7 leaves
c○: Michael Kohlhase 229
We will now establish a few properties of these balanced binary trees that show that they are good
building blocks for combinatory circuits.
Size Lemma for Balanced Trees
Lemma 398 Let G = 〈V, E〉 be a balanced binary tree of depth n > i, then the set
Vi := {v ∈ V | dp(v) = i} of nodes at depth i has cardinality 2 i .
Proof: via induction over the depth i.
P.1 We have to consider two cases
P.1.1 i = 0:
then Vi = {vr}, where vr is the root, so #(V0) = #({vr}) = 1 = 2 0 .
P.1.2 i > 0:
then Vi−1 contains 2 i−1 vertices (IH)
P.1.2.2 By the definition of a binary tree, each v ∈ Vi−1 is a leaf or has two children that
are at depth i.
P.1.2.3 As G is balanced and dp(G) = n > i, Vi−1 cannot contain leaves.
P.1.2.4 Thus #(Vi) = 2 · #(Vi−1) = 2 · 2 i−1 = 2 i .
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