The question may not be capable of a definitive answer under existing Washington jurisprudence. Remarkably, despite how commonplace this situation likely is, the law is not well developed in this area. In particular, this factual scenario appears to involve the intersection of two different legal principles.
First, as indicated by others in this post, it is generally recognized that a property owner may, in most circumstances, cut encroaching branches and roots (only up to the property line) from trees originating from the adjoining owner’s property. In other words, your neighbor could likely trim the branches and roots of the tree, and there would likely be no legal recourse.
Second, one of the three intermediate appellate courts in Washington has held that “a tree, standing directly upon the line between adjoining owners, so that the line passes through it, is the common property of both parties, whether marked or not; and trespass will lie if one cuts and destroys it without the consent of the other.” See Happy Bunch, LLC v. Grandview North, LLC, 142 Wn. App. 81, 93 (2007). In its holding, the Court expressly rejected the argument that “absent a showing of an agreement to the contrary, a boundary line tree belongs entirely to the party on whose land the tree was originally planted, with damages calculated accordingly.” In other words, the fact that your tree was originally planted on your property may not be determinative of whether you own it in its entirety under Washington law.
Critically, the Happy Bunch court did not define “tree.” However, in light of the first principle above, the Court was likely referring to the trunk of the tree, because cutting overhanging limbs and roots is generally recognized as lawful. Ultimately, the Court awarded damages to the plaintiffs, because the defendant did cut trees that were “common property” (i.e., straddling the property line). However, the damages were pro-rated based upon the percentage of the trunks on the plaintiff’s property prior to cutting. Interestingly for your purposes, the damages in that case were based upon the “Trunk Formula Method,” which arborists typically use to value trees, with the diameter of the tree being measured at diameter breast height (DBH), which is usually several feet above ground level, with the purpose of measuring above any taper or flair to the tree.
The bottom line is that you should tread carefully. As indicated by others, you could be liable for treble damages, and even attorneys fees or other costs, if the tree is considered “common property” and it was removed without your neighbor’s consent. See RCW 64.12.030; RCW 4.24.630. However, even if the tree is common property, and you elected to remove it, and your neighbor decided to pursue the matter, a court might only award damages in proportion to the percentage of the tree on your neighbor’s property. Based upon your description, it sounds like little to none of the tree would be on the neighbor’s property, depending upon whether the trunk is measured at DBH or at the root taper. Also, if you do decide to remove the tree, you would need to do so without creating other damage to your neighbor’s property. In this case, stump removal would likely be a bad idea. Finally, it is also possible that if your neighbor’s conduct kills your real estate transaction, you might have claims 1) to establish ownership of the tree, and/or 2) for tortious interference with the transaction.