Why does Bud think ideas are like trees?

The answer to this question can be found in chapter 9. Technically speaking, Bud does not say that ideas are like trees. He says that ideas are like seeds.

IT'S FUNNY HOW IDEAS ARE, in a lot of ways they're just like seeds.

The idea and seed comparison comes up throughout the novel, but chapter 9 is the location where Bud explains the comparison. Bud says that both ideas and seeds start out small, and they both are capable of growing into something that is unfathomably larger than the original seed.

Both of them start real, real small and then . . . woop, zoop sloop . . . before you can say Jack Robinson they've gone and grown a lot bigger than you ever thought they could.
If you look at a great big maple tree it's hard to believe it started out as a little seed.

Bud goes on to explain that the idea of Calloway being his father started out just like that little maple seed. Bud explains that the initial seed of the idea was so small that it was likely to blow away with the first little breeze. It's fitting, then, that the idea itself began with a paper flyer of Calloway. That is an item that quite literally would blow away in any kind of breeze. Eventually, the idea that Calloway is his father is the only thing that fills Bud's entire field of vision. The idea is massive and powerful like a huge maple tree.
Once Bud finds Calloway and the rest of the band, he begins to feel accepted, and another seed gets planted, which quickly moves Bud from laughing to crying in an instant:

but sometime whilst I was sitting in the Sweet Pea another seed got to sprouting . . .

He has finally come home and found a loving family.

Bud thinks that ideas are like trees because they have the potential to grow, just like trees grow from tiny seeds.
To Bud, an idea is like a seed. It has the potential to grow into reality. He relates that the idea of Herman E. Calloway being his father began when he first spied one of Herman E. Calloway's flyers. That small idea soon burst "out of the dirt" when Billy Burns teased the boys at the Home. 
Bud remembers how Billy teased him and his friends, challenging them to reveal who their parents were. Billy Burns bet a nickel that none of the boys could reveal who their fathers and mothers were. Upon hearing this, Bud told Billy that his father was Herman E. Calloway. Bud maintains that this is how the idea of Herman E. Calloway being his father was born. That idea then grew and became the impetus for him to travel across the state of Michigan. So, to Bud, ideas are like trees because they have the potential to grow into reality.