Minimum number of rounds to notify a hierarchy of people with DP
I do not think it is correct.Please consider this example:
Suppose $a$ has $k$ children. Two of them($b,c$) are chains of $k+1$ nodes the other are leaves.
We have:


$OPT(b) = OPT(c) = k$
All other $OPT$ are zeroes.


According to your algorithm we should have $OPT(a)=1+k$
But it is not correct:

let order of notifitions be $b,c,.....$ 

($c,b..$ is the same and the other are suboptimal)


Notification for the subroot of $b$ will stop at $0+(1+k)=1+k$ (time=0)
Notification for the subroot of $c$ will stop at $1 + (1+k)=2+k$ (time=1)
We don't really care about other leaves(they will all be notified by $k$)


This is different than the output to your algorithm.
The approach that FrankW proposed is the correct one!