I doubt that the police will file charges given the information you have provided. Even if your neighbor told you he COULD, that doesn't mean that he WOULD or DID. You yourself expressed by stating that you quote believe quote.
If the tree doesn't die, then that is a second problem with pressing criminal mischief charges. What is the damage? That is what the basis of the charge level would be.
I suggest, if you think he may try something of the same nature again, you set a recording instrument activated by movement to record action.
If he is complaining that perhaps your limbs come over on his property or your roots are causing him foundation issues, it is within his right to cut the limbs, and/or the roots at the property line (which could kill the tree lawfully.)
A person commits criminal mischief if he intentionally or knowingly damages or destroys the tangible property of the owner. Texas Penal Code Section 28.03(a). If the amount of the loss is less than $50, it is a Class C Misdemeanor; if the damage is more than $50 but less than $500, the offense is a Class B misdemeanor; and if the damage is more than $500 but less than $1500, it is a Class A misdemeanor.
Class C Criminal Mischief is punishable by fine only, up to $500. Class B would carry up to 6 months in the County Jail and a $2,000 fine. The judge can put someone convicted of a Class B or Class A misdemeanor on probation from anywhere from 6 months to 2 years. Regardless of the degree, you can always ask that the prosecutor make restitution part of any negotiated deal (but they are not required to do what you ask for; they will listen to you though). A judge can also make restitution to you a condition of probation. Make sure you give the police good contact info. They will pass this onto the County Attorney's Office, who will in turn contact you about the damages you suffered.
(If you do not get restitution from the suspect, there is also a Crime Victim's Compensation Fund that you may be able to get some money from. Talk to the County Attorney's Office.)
You may also have a civil remedy by bringing suit in small claims court. Here in Texas, that would be your local Justice of the Peace. You won't need a lawyer if your claim is below $10000 and the local staff will help you fill out the complaints.
I actually once represented a man charged with criminal mischief for poisoning his neighbor's tree. Our client didn't get along well with his neighbor and had a few beers one night and went over into her yard and poured gasoline onto the roots of her tree. The grass died in a circle around it, but the tree itself seemed to suffer no damage. Because the tree was unharmed, my client was innocent of the charge and his case was dismissed. Perhaps you should hire a tree specialist to take a look at the holes. Document what you believe has happened and stay safe.