 Solution

Use the state order Tall, Medium, Short and 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 isEvery 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.