Galton-Watson Generations and Family Trees

Populations Built One Generation at a Time

A branching process describes a population in which individuals reproduce and their descendants may reproduce in turn. Its defining perspective is generational: the individuals alive in one generation produce the next generation, and then the same reproduction rule applies again. A family tree records the same process in a different form, showing who is descended from whom. These views answer different questions: generation sizes describe how large the population is at each stage, while the tree preserves individual ancestry.

Work on a probability space $(\Omega,\mathcal{F},\mathbb{P})$. Generations are indexed by $n=0,1,2,\ldots$, and the population size $Z_n$ takes values in the state space $\{0,1,2,\ldots\}$. We begin with one ancestor, so $Z_0=1$. The number of children of an individual $u$ is its offspring count, denoted $\xi_u$; it is a finite, nonnegative integer. The offspring law is a probability distribution $(p_k)_{k\ge0}$, where $p_k=\mathbb{P}(\xi_u=k)$, $p_k\ge0$, and $\sum_{k=0}^{\infty}p_k=1$.

Definition: A Galton–Watson branching process starts with one ancestor and has the following reproduction rule: every individual has an offspring count with the same law $(p_k)_{k\ge0}$, and the offspring counts of distinct individuals are independent. In particular, conditional on the reproduction history through generation $n$, the individuals in generation $n$ reproduce independently according to that common law. The generation size $Z_{n+1}$ is the total number of children produced by the $Z_n$ individuals in generation $n$.

“Same law” means that each individual has the same probabilities of having $0,1,2,\ldots$ children. It does not mean that their realized offspring counts are equal. “Independent” means that one individual’s offspring count does not determine another individual’s count; it does not mean that the population sizes $Z_n$ are independent across generations. Indeed, $Z_{n+1}$ is determined by the offspring counts in generation $n$, and the number of individuals in a later generation depends on what happened earlier.

From Offspring Counts to a Family Tree

To draw a family tree, put the initial ancestor at the root. Its children are placed at level 1, their children at level 2, and so on. A convenient way to distinguish individuals is to label the root by the empty sequence $\varnothing$. If an individual has label $u$ and has $r$ children, label those children $u1,\ldots,ur$, where appending a number means adding one more entry to the label. For example, the label $23$ identifies the second child of the individual labelled $2$. The order of the child labels is only a bookkeeping convention; it does not change the population size or the reproduction law.

The level of an individual is the number of entries in its label: the root has level $0$, its children have level $1$, and an individual labelled $23$ has level $2$. Thus the individuals at level $n$ are exactly those in generation $n$, and $Z_n$ is the number of vertices at that level. Every vertex except the root has exactly one parent, obtained by deleting the last entry in its label. This parent-child relation is what makes the representation a tree.

Theorem: In a family tree with one root and finite offspring counts, the vertices at level $n$ are exactly the individuals in generation $n$. If generation $n$ is finite, then $$ Z_{n+1}=\sum_{u\text{ in generation }n}\xi_u. $$ Every generation is finite, and if some generation has size zero, all later generations have size zero.

Proof. The root is the unique individual in generation $0$ and is the unique vertex at level $0$. Suppose that the individuals in generation $n$ are exactly the vertices at level $n$. Each such individual $u$ has precisely $\xi_u$ children, and the labels of those children are formed by appending one entry to $u$. They are therefore at level $n+1$, and every vertex at level $n+1$ is obtained from its parent in this way. This proves the level-generation correspondence by induction and shows that the number of vertices at level $n+1$ is the sum of the children counts in generation $n$. For finiteness, generation $0$ has one member. If generation $n$ is finite, it is a finite set of individuals, each with a finite number of children, so their total number of children is finite. Induction gives finiteness of every generation. Finally, if generation $n$ has no individuals, there are no parents to produce generation $n+1$, so that generation is empty. Repeating the same argument shows that every later generation is empty. $\square$

The displayed sum is a pathwise counting identity: it holds for each realization of the family tree. The independence and common-law assumptions specify how likely different offspring counts are, but they are not needed to count the children once the counts are known. This distinction is useful: first identify the individuals and their offspring counts, then use the probability model to assess the chance of that realization.

Worked Example: Every Individual Has Two Children

Suppose the common offspring law assigns probability $1$ to two children: $p_2=1$, and $p_k=0$ for every $k\ne2$. Begin with the root $\varnothing$. It has children $1$ and $2$, so $Z_0=1$ and $Z_1=2$. Each of those two individuals has two children, giving the labels $11,12,21,22$ and $Z_2=4$. Continuing once more gives eight individuals at level 3.

Z_0=1,\qquad Z_1=2,\qquad Z_2=4,\qquad Z_3=8.

At each step the sum identity gives $Z_{n+1}=2Z_n$, because each of the $Z_n$ individuals has exactly two children. Substitution checks the first three transitions: $2\cdot1=2$, $2\cdot2=4$, and $2\cdot4=8$. Since every offspring count is $2$ with probability $1$, this sequence occurs with probability $1$.

Independence Applies to Individuals

When offspring counts are random, it is important to keep track of whose offspring count is being specified. In the family-tree labels, each individual has its own count. The common-law assumption gives the same probability to a specified count for every individual; independence lets us multiply probabilities for a finite list of specified counts. The list must include only individuals who exist in the realization being described. An individual cannot have an offspring count in the tree before being born.

Worked Example: Different Siblings Reproduce Differently

Let every individual have either $0$ or $2$ children, each with probability $\tfrac12$, independently of all other individuals. Consider the following realization through generation 2. The root has two children, labelled $1$ and $2$. Individual $1$ has no children, while individual $2$ has two children, labelled $21$ and $22$. Then individual $21$ has two children and individual $22$ has no children.

The generation sizes follow directly from the labels. Generation 0 contains $\varnothing$, generation 1 contains $1,2$, generation 2 contains $21,22$, and generation 3 contains $211,212$. Thus $Z_0=1$, $Z_1=2$, $Z_2=2$, and $Z_3=2$. In particular, two siblings can have different realized offspring counts even though they share the same offspring law.

For this exact sequence of specified outcomes, the five relevant individuals are the root, $1$, $2$, $21$, and $22$. Their specified offspring counts are respectively $2,0,2,2,0$. Each specified count has probability $\tfrac12$, and the counts are independent. Therefore the probability of this realization through the reproduction of generation 2 is $$ \tfrac12\cdot\tfrac12\cdot\tfrac12\cdot\tfrac12\cdot\tfrac12=\tfrac1{32}. $$ There are no other individuals in generations 0, 1, or 2 whose offspring counts need specifying. Checking by the counting identity gives $Z_1=2$, $Z_2=0+2=2$, and $Z_3=2+0=2$, in agreement with the labels.

The independence assumption does not say that generation sizes are independent. In the example, the event that generation 1 contains two individuals is determined by the root’s offspring count. Generation 2 is then formed from the offspring counts of those particular individuals. The set of individuals whose counts contribute to a generation depends on the earlier generations. The model is independent at the level of distinct individuals’ reproduction, while the population sequence is linked across time by descent.

A Mixed Offspring Law and Its Tree

A common offspring law may assign positive probability to several possible family sizes. The same rule still applies to every individual, and the tree records the realized outcomes rather than replacing them by an average. For a particular finite realization, its probability can be computed from the offspring outcomes of the individuals that have already appeared, using the common probabilities and independence.

Worked Example: Three Possible Family Sizes

Suppose the offspring law is $p_0=\tfrac12$, $p_1=\tfrac13$, and $p_3=\tfrac16$, with $p_k=0$ for all other $k$. These probabilities sum to $\tfrac12+\tfrac13+\tfrac16=1$. Consider the event that the root has three children, and that these children, in label order, have $0$, $1$, and $3$ children.

The root’s children are labelled $1,2,3$. Individual $1$ contributes no children, individual $2$ contributes the child $21$, and individual $3$ contributes $31,32,33$. Hence generation 1 has size $Z_1=3$, while generation 2 has size $Z_2=0+1+3=4$. The four vertices at level 2 are $21,31,32,33$.

The root outcome has probability $\tfrac16$. Conditional on the root having produced these three individuals, the probability of their specified offspring counts is $\tfrac12\cdot\tfrac13\cdot\tfrac16=\tfrac1{36}$. By independence, the probability of the full specified event is $\tfrac16\cdot\tfrac1{36}=\tfrac1{216}$. The count check $Z_2=\xi_1+\xi_2+\xi_3=0+1+3=4$ agrees with the four listed labels.

Extinction and a Common Counting Error

In this generation-by-generation description, extinction means that the population is empty in some finite generation: there is an $n\ge0$ with $Z_n=0$. Because an empty generation has no individuals to reproduce, the process remains empty thereafter. This is a statement about the structure of the model, not a claim that extinction must occur. A tree may instead have individuals at every level. If that happens, it contains infinitely many individuals, since the levels are distinct and each has at least one member.

A frequent mistake is to confuse the offspring count of one individual with the size of the next generation. For example, if a generation has three individuals with offspring counts $0,1,3$, it is incorrect to set the next generation size equal to $3$ merely because one individual has three children. The correction is to add the counts for every individual in the current generation: $0+1+3=4$. The family-tree labels provide a direct check: one child comes from the second individual and three from the third, giving four vertices at the next level.

Another mistake is to assume that identical offspring laws force identical family sizes. The law describes probabilities, not a fixed outcome. Under the law with probabilities $\tfrac12$ for $0$ and $\tfrac12$ for $2$, one sibling may have no children while another has two. Their outcomes differ, but each follows the same distribution. Independence concerns the joint probabilities of these individual outcomes; it does not erase the dependence of later population sizes on the tree already formed.

The two representations are complementary. Use $Z_n$ when the question concerns the number of individuals in a generation. Use labels and the family tree when the question concerns ancestry or which individuals produced a particular generation. In either representation, the central bookkeeping rule is that each individual in generation $n$ contributes exactly its own offspring count to generation $n+1$.

Check Your Understanding

Use the generation-counting identity and the family-tree labels to answer these questions.

If $Z_n=4$ and the four offspring counts are $1,0,2,1$, what is $Z_{n+1}$?
Under a law assigning probability $\tfrac12$ to $0$ children and $\tfrac12$ to $2$, can two individuals in the same generation have different offspring counts? Explain.
For the offspring law $p_0=\tfrac12$, $p_1=\tfrac13$, $p_3=\tfrac16$, what is the probability that a specified individual has exactly one child?
Why does $Z_n=0$ imply $Z_{n+1}=0$?
If a root has children labelled $1$ and $2$, and individual $2$ has child $21$, which generation contains $21$?