Lexers eat streams of characters that they digest into tokens. Tokens are like food and water for compilers.

Parsers - Syntax Analyzers - produce trees and sometimes forests. And many of those trees very much resemble flowers.

If you are not careful enough while designing your grammar your trees grow wild.

Later, during semantic analysis you decorate them with attributes, very much like a christmas tree.

And much later - during machine independent optimization phases - you have to transform their shape, pruning them, forcing them to be the way you want (a bit like bonsais).

Finally during code generation and machine dependent optimization bees and insects visit the trees producing the final translation.

But at the end I feel that general purpose programming is more like writing mathematics.

Each time you write a correct program (exists such thing?) you are attempting (via reasoning, verification and testing) to prove that your input-output function is the one your client requires. You are writing, somehow, a theorem.