= Solution
Use the state order Tall, Medium, Short and \b[column <probability distributions>], so that $\mathbf p_{n+1}=P\mathbf p_n$. The $j$th column records the child's <probability distribution> given parent state $j$. Thus the <transition matrix> is
$$
\boxed{P=\begin{pmatrix}
0.6&0.2&0.1\\
0.2&0.7&0.3\\
0.2&0.1&0.6
\end{pmatrix}.}
$$
Every entry is nonnegative and every column sums to one. This convention also gives the requested property $P^T(1,1,1)^T=(1,1,1)^T$. A row-distribution convention would use the transposed <transition matrix> consistently instead. The stated <Markov chain> is the mathematical model to analyze; no additional biological assumptions are needed.
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