My gutter downspouts are connected to storm drain by in-soil 4-inch
corrugated hoses. One of these hose has small tree roots in it about 5 feet
into the opening. The hose curves towards horizontal so it is not direct
line of sight; I snake in a bullet cam to see it.

It still drains water, but if I stick a garden hose in it and turn on water
full throttle, it backups and overflows after some 10 seconds or so, which
seems to imply there are more tree roots down deeper into the hose (it takes
less than 10 sec to overflow 5 feet of hose).

I can tolerate the current drain rate since rain water is not sufficient to
overflow the hose. But how do I prevent it from getting worse?

There are some bushes and a douglas fir within 8 feet of the hose opening. I
don't know whose root it is.