Bsrgvty

Maybe, nature is incredibly inventive, but there are a couple of problems with the basic premise;

1. Glaciers "bulldoze" soil out of the way when they move down valley. If there is any major volume of ice around the pre-freeze soils have been pushed off the slopes.




2. The glaciers aren't going to stop moving for the trees that are growing through them either. The life expectancy of anything growing through an active glacier is measured in months; the time between first frost (when the glacier freezes to it's bed for the winter) and the thaw (when it starts to move for the spring and summer months). In temperate areas that can have winter rains it's even worse as winter storms can cause short melt spells, and glacial surges, in the otherwise stable cold months.




3. They don't have to; trees, in fact whole forests, can and do grow on top of glaciers, soils often form on the lower sections of ice flows, supporting a variety of plant life. I myself have seen willow and poplar growing on soil covered ice near the toe of Fox Glacier in New Zealand and there are pine forests on the lower lobes of a couple of European glaciers.

As for roots melting through the ice chemical rather than thermal melting is probably easier to explain, salt excreted from the roots could raise the freezing point sufficiently to thaw down through the ice, water absorption would be automatic due to osmotic pressure.

Up from below.

https://morgan-outdoors.com/?p=1443

Your glacier trees can take some pages from the playbook of the invasive Japanese knotweed, famous for its ability to break its way up through concrete.

1. Your plants establish on the edge of the glacier. Runoff provides liquid water. The soil is good. Organic nitrogen is in short supply but these plants can fix their own nitrogen.




2. They put down deep roots. Knotweed roots are 5 meters down. The soil next to a glacier facilitates this.




3. They spread out in a clonal patch; if these are trees then like an aspen clone. Plants come up thru the adjacent ice.




4. As new roots are laid down deeper into the ice field, the plant can encroach its way into the ice. Ice is broken as it goes. If this just an ice field, the plant will facilitate the dissolution of the ice. If this is a flowing glacier then it will fight back.

This sidesteps the issue of nitrogen and puts your plants over and under the ice.

Last piece: slider seeds.

These are big seeds, like coconut. They have enough resources in each seed to establish a hardy well rooted plant on minimal resources, like the coconut does for a sandy beach. These big seeds have sails and slide or tumble across the ice fields until they hit a low point and lodge. If this is a fissure or a bare patch or somewhere with access to the soil then the tree can get established and start a new clone.

Maybe instead of just seeds, during storms entire trees can break off from the clone and slide across the ice. Knotweed can definitely break off and take root in a new place from loose fragments with associated resources. . With access to water it can cannibalize its parts to grow the deep root needed to establish itself deeper in the ice field.

Fertile soil is fertile, because it contains nutrients, and nutrients come from decomposed organic matter. If the place is covered in ice for 3mln years, and all organisms which live there live on the top of the glacier, I'm afraid the soil beneath it can't be fertile.
Unless there is something else going on, like for example another ecosystem of organisms living under the ice. But in that case you would need much more than three million years for the evolution to make it work.

Back to your main question: Ice thickness on Antarctica, as well as thickness of glaciers during the most recent ice age, was about 2-4km. Trees growing 2-4km long roots from a single seed would be very difficult to rationalize, since only after touching the fertile soil beneath the tree would be able to get additional resources to build itself.
Maybe you can say that the seed builds a carbon nanotube, drills it through the ice, and then sucks in nutrients and water. But again, three million years is not enough to evolve something like that. To be honest, I don't think any number of years would be enough - your readers will think you're just hand-waving things. Instead, maybe, the "seeds" could in reality be complex nanomechanisms, planted on the planet by an advanced civilization. The trees that grow out of them are real trees in every aspect except for that their own new seeds are really built by nanobots living inside them.

I hope that helps.

Lightning trees!

The first law of living in extreme environments is: never expend resources you don’t have to. In your case, why do hard work getting down to the soil if you can get a lightning storm to do it for you?

Assuming that some part of the year, the surface temperature is above freezing (or you can’t have plant life anyways) your glacial plateaus will have lightning storms, with liquid water coming down, to boot.

Ice is a terrible conductor, and your trees are a tall and jutting path to ground. Cue:

“Ah!” but you say, “But then my tree has exploded!”

Well, that’s okay. That’s exactly what your tree wanted! It’s an active pyrophyte.
https://en.m.wikipedia.org/wiki/Pyrophyte

So it explodes, and sends chunks of itself skittering around the whole damn surface of the glacier! Maybe like eucalyptus trees, its entire canopy can detach and get blown miles away... where it attracts a second lightning strike, which cracks and melts a path down to the soil beneath, where your pyrophyte seeds happily settle in to grow in newly wet soil.

The main issue with roots going all the way through the glacier is that glaciers move.

More importantly, they move through deformation. This means that the base of the glacier is moving at a different speed than the surface. A solid object going all the way through the glacier (e.g. root) will get stretched, torn, and destroyed.

Many other answers provide alternatives that would still allow you to have a glacier covered in forest.

Not all glaciers are sparkling white. Some are dirty, in that they contain a certain amount of soil. The soil could be churned up from the ground the glacier flows over and rises to the top, or it can be deposited by other means, such as wind storms or volcanic eruptions.

This soil could be the source of your nutrients in the same way hydroponics works. Granted, the nutrients won't be in large amounts nor will they be quickly replenished, but you have 3 million years worth of soil buildup as well as old trees dying and decaying.

If there is any fauna, that would add nutrients as well. This would be due to their dragging their own edibles onto the ice, manure, urine, or their own corpse.

Because of the minimal nutrients, you're likely to see longer lived trees rather than massive forests. Even on Earth, there are trees that are hundreds, if not thousands, of years old. On this planet, they might be tens of thousands of years old. They may also be really massive, or relatively small for their age. Their growth rings may also be so small as to not really be distinct enough to be rings anymore.

Once a tree dies, it leaves it's nutrients to it's descendants, so maybe there are pockets of trees as well as large individual trees.

Could trees adapt to grow down through the ice and snow to reach the fertile soil beneath?

Plant, like any other form of life we know on Earth, depend on liquid water to perform their biological functions.

The main issues with a frozen planet surface is that it lacks liquid water, and that the low temperature really slows down biological chemical reactions.

Moreover, I suspect that after 3 million years of ice age the ice sheet would be rather thick, too thick for whatever root to pierce it starting from the top.