No real mystery here.

For an "ordinary" puckered lip whistle, the pitch is determined - like a "Helmholtz Resonator" - by the volume of air immediately behind the "pucker," divided by the cross section area of the hole(s) connected to that volume.(There's at least one hole where the air comes in, and another where it goes out.)

This is the same "resonance chamber" you use when singing, so the frequency range will probably be about the same.

When singing, you can "go falsetto" by driving your vocal chords to a harmonic of the natural frequency of the air volume. It is rather difficult - with air flow only - to make a Helmholtz volume "break" to a harmonic, so with unsophisticated whistle technique, you probably can't whistle in your "falsetto" range.

Blowing through the normal pucker, you can change pitch by shoving the tongue forward and, by various roll, curl, and tuck techniques, creating a smaller volume immediately behind the pucker. Note that the "inlet" hole counts in determining the pitch too, so you have to develop a technique for minimizing it too, while keeping an "open-but-small" resonance chamber much smaller than your whole mouth.

As mentioned, it is very difficult to "overblow" to a harmonic with a Helmholtz chamber, but a few people are able to form a "long-slender" air pocket with the tongue that can produce overtones - although it tends to be sort of a "falsetto whistle."

A "slit lip" whistle, where the lips are pulled back as in a big grin (grimace) can use the small air pocket between the bottom of the tongue and the lower teeth to produce very high pitched (and loud) notes, but usually with only a very limited pitch range. Useful for signalling your hound, but not much valued in musical endeavers.

The "two fingers in the mouth" method can also be used to produce a very small air volume and hence a high pitch, but most people who use this method are literally "one note Johnnies."

The imponderable part of this is - since the development of versatile and precisely controllable tongue thrusting is essential to being a first class warbler, will bottle feeding our children hinder (or help) their musical careers?

John