TL;DR: Nature favors packing nucleons in pairs of anti-parallel spins. (electrons as well!)
The zig-zag is a nucleosynthesis artifact.
H, He and Li are "primordial" (made out of the abundant neutrons in the first hours after the Big Bang.)
Everything else is made in the stars.
There are a few main processes that make heavier elements (oversimplified!):
- Hydrogen burning (makes helium as we already don't have great deal of it)
- Helium burning (makes C and O)
- Helium (even element itself) + any even element (makes even elements up to Fe)
- Slowly adding neutrons to everything and waiting for it to beta-decay (called s-process, makes odd elements out of even elements and elements heavier than Fe, up to some point)
- Rapidly adding neutrons without waiting much anything to beta-decay before adding the next neutron (r-process, makes all elements up to uranium out of existing elements from previous processes)
Every one of these processes gets its material from the previous ones and has less yield.
That's why we have heavier elements less than lighter elements, even elements more than odd elements and r-process elements less than s-process elements.
Edit: Both s- and r-processes, as well as other ones not listed here, leave ultimately more even elements as they are in general more stable against both beta-decay and capturing the next neutron. Why they are more stable? Because both protons and neutrons fill their very own shell-like orbitals inside the nucleus and nature favors packing nucleons in pairs of anti-parallel spins. (electrons as well!) See Nuclear shell model