Foundations of Data Science

Theorem 4.11 Consider a tree generated by a branching process. Let f(x) be the generating
function for the number of children at each node.
1. If the expected number of children at each node is less than or equal to one, then the
probability of extinction is one unless the probability of exactly one child is one.
2. If the expected number of children of each node is greater than one, then the probability
of extinction is the unique solution to f(x) = x in [0, 1).
Proof: Let p i be the probability of i children at each node. Then f(x) = p 0 + p 1 x +
p 2 x 2 + · · · is the generating function for the number of children at each node and f ′ (1) =
p 1 + 2p 2 + 3p 3 + · · · is the slope of f(x) at x = 1. Observe that f ′ (1) is the expected
number of children at each node.
Since the expected number of children at each node is the slope of f(x) at x = 1, if
the expected number of children is less than or equal to one, the slope of f(x) at x = 1
is less than or equal to one and the unique root of f(x) = x in (0, 1] is at x = 1 and the
probability of extinction is one unless f ′ (1) = 1 and p 1 = 1. If f ′ (1) = 1 and p 1 = 1,
f(x) = x and the tree is an infinite degree one chain. If the slope of f(x) at x = 1 is
greater than one, then the probability of extinction is the unique solution to f(x) = x in
[0, 1).
A branching process with m 1, then the branching process will die out with some probability less than one unless
p 0 = 0 in which case it cannot die out, since a node always has at least one descendent.
Note that the branching process corresponds to finding the size of a component in
an infinite graph. In a finite graph, the probability distribution of descendants is not a
constant as more and more vertices of the graph get discovered.
The simple branching process defined here either dies out or goes to infinity. In biological
systems there are other factors, since processes often go to stable populations. One
possibility is that the probability distribution for the number of descendants of a child
depends on the total population of the current generation.
Expected size of extinct families
We now show that the expected size of an extinct family is finite, provided that m ≠ 1.
Note that at extinction, the size must be finite. However, the expected size at extinction
could conceivably be infinite, if the probability of dying out did not decay fast enough.
To see how the expected value of a random variable that is always finite could be infinite,
let x be an integer valued random variable. Let p i be the probability that x = i. If
∞∑
p i = 1, then with probability one, x will be finite. However, the expected value of x
i=1
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