"...Maybe you have some redemption to offer?"
IF i do(and i just hate to give ground to Such scum...),it'd be this,a fairly simple expedient:I'd relate that symbolism of a Tree to the ancient custom of tacking a living green branch to the gable of a freshly-completed timber-frame.
And this,in turn(without getting too terribly romantic and evoking Druids et c.

),i'd simply credit to a simple,reasonable and most natural Appreciation of the Woodworker(carpenter,cabinetmaker,joiner,et c.) for this absolutely essential to all of us Material-WOOD.
Tree growth,greenery,it's been so downright Handy for us humans,what could be more natural than to decorate the woodworking tools with symbolism of trees growing and blooming and perpetuating themselves?!
Thanks for the photos of all these lovely tools,neat stuff.And i agree with 300Six in his analysis of the specifics of elm structure,sure does seem like an excellent choice per application.
A silly historical side-note:A bit over 1000 years ago,these two Byzantine monks,Cyrill and Mefodii,have worked up what is now known as "Cyrillic" alphabet,to teach literacy to a bunch of new converts speaking whatever Slavic dialects.
Along with that they developed the simple methodology of teaching it,where the visual shape of the letter was associated with a concept,like on childrens' wooden blocks,B=Bee,et c.
And later,the words that kept evolving reflected that an whatever degree.
One of the letters that they made up(cyrillic has 33 letters) was Ж.It took it's shape from the "dendritic",or tree-like,branchy pattern.And the symbol- word for it became at first Живот(belly,or stomach),later transforming into Жизнь (life

)
Yet later it caused a logical consequence where the old style external iron hinges on doors got the name Жиковина,for their obvious sprawling,tree-like visual appearance.
And those,located as they were on the main opening to the dwelling(where the evil spirits were most likely to penetrate

),of course symbolised the Benign,Life-affirming influence of trees and all things green and wonderful.....
