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Since air is a newtonian fluid, so we can use the newtonian law for the force of friction (1)
Since we don't want to work with differentials here (and we can't), we'll approximate it relatively well using the quotient of the velocity and the boundary layer thickness. 
Boundary layer thickness will be a real pain in the ass, so we do what physicists do best: 
Approximate the shit outta that thing. 
If we assume the sword to be a flat plane (so, ignoring the angle at the blade), go into a coordinate system in which the blade is resting, thus having a freestream velocity equal to the blade's velocity, also assuming the maximal length of shearing the fluid can do is our path per cut and that out viscosity is constant (which it is NOT in real life, but we really don't want to turn this into a high-grade differential equation) we can use equation (2). 
(A simplified version of the approximated boundary layer thickness following the Blasius solution.)

Okay, so now we can calculate the Force of friction. 
Now the real approxi-fun begins. 

Krieger has to put constant work into the sword to cancel out this friction so the sword stays the same speed (since friction tries to slow it down). 
Work is a simple equation. Also, Energy is basically work. And friction, like the little bitch it is, converts all the work into heat. 
Now we'll just say that 70% of this heat goes into the sword and 30% into the sourrounding air. 
(Normally it would be more like 50/50, but at supersonic speeds, the ratio changes to the heat-isolated site.) We'll also up that number, because, you know, wood probably has a bit more friction than air and the sword is rubbing across that poor tree like a motherfucker. 
For this, which I have to shoehorn a bit, I'll take the inverse value of the steel-wood friction coefficient (~1.7), which is most probably way too low, but who cares. 
So we now have an equation for the thermal energy of the sword, (3)

And NOW we start to get idiotic, because if we assume that the sword is actually an ideal atomic gas, we can calculate it's temperature from the thermic energy it has. So, if our result is just plain stupid, this is probably the reason why. 
Anyways, we'll continue with it. (4)

Getting close to the end. Who am I kidding, nobody will read this anyways. But it's fun and good practice for me, so fuck you. 
We're now putting those 4 equations together to get a long, unecessary complex equation for the temperature the sword gains through the supersonic tree-cutting movement.  (5)
Which I'm too lazy to simplify further, I'll just punch the numbers in it.