5. A certain public figure and all of his descendants, regardless of gender, are included in the family tree, but their spouses are not included in the family tree. The family tree only indicates whether a member is male or female and the parent-child relationships between them. It is known that the family tree of this public figure includes a total of $n$ male and female members, each member has at most two children, and if they have two children, they must be one male and one female. How many possible family trees are there for this public figure?


4. The family tree of a certain public figure can be represented by a tree diagram similar to the one on the right. This family tree can be constructed as follows: each member is represented by a black dot, and those with father-son (or daughter) and mother-son (or daughter) relationships are connected by line segments; any member's son is drawn below and to the left, and the daughter is drawn below and to the right. Let $a_{n}$ be the number of such family trees, clearly $a_{1}=1, a_{2}=2$, and let $a_{0}=1$. Now consider $n \geqslant 3$, if we remove the certain public figure from the family tree, it becomes two family trees of the public figure's son and daughter. There are the following possible scenarios: the son's lineage has $k$ people, and the family tree has $a_{k}$ possibilities, at this time the daughter's lineage has $n-k-1$ people, and the family tree has $a_{n-k-1}$ possibilities, i.e., there are $a_{k} a_{n-k-1}$ possibilities, where $k=0,1,2, \cdots, n-1$. Thus, we have the relation $a_{n}=a_{0} a_{n-1}+a_{1} a_{n-2}+\cdots+a_{n-1} a_{0}$.
Let the generating function of the sequence $\left\{a_{n}\right\}$ be $f(x)=\sum_{k=0}^{\infty} a_{k} x^{k}$, then
$$
\begin{aligned}
a_{0}=a_{1} & =1, x \cdot f^{2}(x)=x\left(a_{0}+a_{1} x+a_{2} x^{2}+\cdots+a_{n} x^{n}+\cdots\right)^{2} \\
& =a_{0}^{2} x+\left(o_{0} a_{1}+a_{2} a_{0}\right) x^{2}+\cdots+\left(a_{0} a_{n-1}+a_{1} a_{n-2}+\cdots+a_{n-1} a_{0}\right) x^{n}+\cdots \\
& =a_{1} x+a_{2} x^{2}+\cdots+a_{n} x^{n}+\cdots=f(x)-1 .
\end{aligned}
$$

Thus, the generating function $f(x)$ satisfies the equation $x f^{2}(x)-f(x)+1=0$. Therefore, $2 x f(x)=1 \pm \sqrt{1-4 x}$, but since $f(0)=a_{0}=1$, we have $f(x)=\frac{1}{2 x}(1-\sqrt{1-4 x})$.

Applying formula 5, expanding $(1-4 x)^{\frac{1}{2}}$, and substituting it into (*) yields (the process is omitted) $f(x)=\sum_{k=0}^{\infty} \frac{1}{k+1} C_{2 k}^{k} \cdot x^{k}$. Hence, $a_{n}=\frac{1}{n+1} C_{2 n}^{n}(n \geqslant 0)$ is the desired result.