  Let $F(p)$, $p\ge2$ be the family of generalized Thompson's groups. Here F(2)
is the famous Richard Thompson's group usually denoted by $F$. We find the
growth rate of the monoid of positive words in $F(p)$ and show that it does not
exceed $p+1/2$. Also we describe new normal forms for elements of $F(p)$ and,
using these forms, we find a lower bound for the growth rate of $F(p)$ in its
natural generators. This lower bound asymptotically equals $(p-1/2)\log_2
e+1/2$ for large values of $p$.
