Q. The tree philodendrum I've had for 15 years is becoming extremely stalky and I wonder if I might cut the brown stalk and re-root the leafy part.
The look of a tree philodendron (Philodendron bipinnatifidum) is less attractive to most people as it gets tall. This article from the University of Arkansas Cooperative Extension provides an interesting portrait of the plant and describes how the plant becomes a climber in its natural habitat.
While propagation is typically from seed, it is possible to propagate from a cutting taken from a vigorous shoot low on the plant. The entire top of your plant is probably too large for a budding set of roots to support successfully (though you can try if you want, reducing the leaves attached to just a few so that transpiration is not too intense).
A three to four inch cutting taken in early summer has a higher chance of success. Cut a low growing tip with a sterile implement below a node and remove the lowest leaves before planting in a moistened peat and perlite potting mixture. If the tip has large leaves, a single cutting should go in a four to six inch pot.
Enclose in a clear plastic bag and seal the bag around the pot to reduce the loss of moisture; place in indirect, bright light at normal home temperatures. Rooting should occur within four weeks and new top growth indicates that the bag can be removed and the plant can be watered and fertilized sparingly. Repot in standard potting soil and treat as an adult plant after three months total from taking of cutting.
Good luck with this project!