What is the meaning of the poem "On Killing a Tree"?

I don't know that there is a single, correct interpretation for a poem like this. As such, I believe it will be most helpful to look at the idea or principle the poem presents, and then see if it is relevant in other contexts apart from trees.

The poem has four stanzas (plus a couple introductory lines) and, coincidentally, also has four elements to it, each expressed by its own stanza. They are: 1) the tree itself, 2) the ineffective attempt at killing the tree, 3) the roots, and 4) the successful killing of the tree.

Working with these, the poem can briefly be summarized as: trees are quite resilient; even chopping them down isn't enough to kill them; as long as the roots survive, so will the tree; trees can only be truly killed by ripping the roots from the ground—the source of sustenance and safety.

With the poem simply summarized, it becomes pretty clear that the idea behind the poem is the durability and capability of living things to recover from even grievous wounds, so long as they have a safe, nurturing "root" intact.

There are a lot of directions you can take this. To me, the ones that jump out are things that could be described as "living" yet aren't organisms in the biological sense—namely, communities or the human spirit.

Communities, much like trees, slowly grow and expand over the years from just a small starting point (stanza 1). Further, removing some members of the community, though often devastating, is not so much that it cannot be recovered from, given time (stanza 2). Because communities are typically founded not by coincidence, but by a group coming to together with shared needs or beliefs, there is usually a root cause (see what I did there?) that holds the community together more than any single member (stanza 3). If one destroys these roots of the community, even if all the members remain, the community will eventually wither and fall apart (stanza 4).

The other interpretation I mentioned, the human spirit, can also be understood with regard to the idea Patel presents in the poem. People have suffered tremendously and needlessly throughout history—I don't imagine I need to give examples—but there are just as many stories of people in those situations that somehow endure and recover (not just physically, but emotionally and mentally as well). Many have scars, just as a once chopped down tree would, but they have recovered nonetheless (stanza 2). To truly break someone, you must destroy their hope, crush their spirit. And that is not as easily done, especially if they have support from others. How much more difficult is it to endure when on your own? This system of nurturing and support is represented by the earth, and our connection to it is the roots (stanza 3). The world can be a terrible place, and without a connection to keep us sturdy and grounded, our spirit is exposed to the elements. And once we are separated, once we are alone:

Then the matter
Of scorching and choking
In sun and air,
Browning, hardening,
Twisting, withering,
And then it is done.

In the same way a tree is dependent on its roots to survive, so too is the human spirit dependent on others for support. The world is too taxing and painful to survive without it (stanza 4).

That turned rather grim. Whoops.

These are, of course, not the only possible answers. But hopefully they're enough to work from and to get an idea of what the author could have been talking about.