I think the way brain cells grow in our brain is much akin to the way branches grow in a tree.
It's not. Neurons, unlike trees, are physically interconnected with one another, and the number and strength of those connections is determined by a combination of hormones and activity. New connections can and do form throughout a person's life while old/unused connections atrophy until the junctions between them separate (this is the process of forgetting
If you have two plants that are the same age and are clones, then each tree grows its branches in different locations, that is a random fractal process
Neuronal growth is not random, nor is it usually fractal. Most neurons -- IIRC -- are monopolar containing a single axon that can connect to one other neuron.
There are unused dendrites that don't presently have connections and are available to accept new connections from neighboring cells. Neurologists have figured out a rule of thumb that "neurons that wire together fire together" and tend to depolarize synchronously. The more neurons connect to the same juncture or the same cluster of cells, the more can fire together to create a stronger impulse.
I'm not saying its entirely random, but our associative memory works that way.
Again, there's nothing random about it. It's a matter of repetition and patterning more than anything else. When two groups of neurons tend to be triggered by the same stimuli all the time, they have a tendency to form connections with one another; later on, different stimuli that would have otherwise only triggered one of them ends up triggering the second one as well, forming an association.
How this work sin memory is that a certain experience happens to you a certain number of times or in a certain way that causes a whole cluster of specific neurons to depolarize all at once. Those neurons immediately strengthen existing connections (or sometimes form new ones), resulting in a relational network. When you encounter a stimuli that's similar to this one, it will trigger SOME of the neurons in that relational network, and the others are triggered by their association, with the network itself.
Some of that stimuli is self-generated within the brain; as I type this, for example, an extremely well-developed cluster of neurons in my parietal lobe are firing like crazy in very specific patterns associated with a very specific spatial location in my parietal eye field. Those connections are so strongly reinforced by now that I am now able to type this message without actually looking at the keyboard; my brain is keeping track of my fingers and has a very good working memory of where the keys should be from the relative position of where my fingers are. This was NOT learned by random action, but by repeating the same action over and over and over and over and over again.
That's the key to human learning. Not randomness, but intensive pattern repetition.