There was an attempt at flanking, actually. The French plan of attack called for two cavalry wings (one for each flank) to charge up the sides of the battlefield and sweep away the English archers, hopefully preventing them from making much of an impact on the main battle. This cavalry attack failed, largely as a result of its lack of coordination. The French were taken by surprise as the English changed positions, so the archers had set up their stakes and were prepared to shoot before the cavalry could organize itself to charge. The cavalry charge resulted in complete failure and numerous casualties.

As for why they didn't go through the woods? Infantry (which formed the bulk of the French army) could theoretically move through the woods, but not in the close order required for a proper attack. If they went through the woods, the English could have easily overwhelmed the scattered line. In fact, that's sort of what happened anyways, because the combination of muddy fields + concentrated archery volleys severely disrupted the main French attack. So instead of one huge push smashing into the English line, there were scattered assaults all along the front that were almost completely unsupported.

During this age of war you need 2 things:

Massed formations of men (infantry/archers) Massed formations of Cavalry

You can't really mass anything in a forest, but the guys at the edge of the forest (aka the english) can mass together perfectly well, and will decimate the loosely packed french trying to come out of the forest and organize, in the mud, before radios, trying to untangle the polearms from the branches, while getting stabbed.

For obvious reasons you can't charge cavalry through a forest, so that's right out.

Additionally, its slower to walk through a forest than over an open field, so if the french go into the forest to attempt a flanking move, the english will simply outflank and surround them by surrounding the forest, since they can move more freely in the open fields.

The forest doesn't need to be very dense. Even if it's sparse, your formations still have to "open" and "close" as they pass around/over the natural obstacles of trees and roots and rocks and whatever. This is a major vulnerability. So major that the french decided a cavalry charge across a muddy quagmire was still better than going through the forest.

Generally speaking skirmishers and raiders were fond of forests, but dedicated set piece battles need lots of open terrain where men can stand shoulder to shoulder as a cohesive unit. It's not specific to Agincourt, it's just how war worked for everything until up until the self contained cartridge.

Much appreciated for clearing that up!