The silver birch has leaves falling, which have some spotting on them. I’m wondering if the tree has some sort of infestation?
This might be just because of the fall period where there is more fungal matter in the air. To be honest, I wouldn't worry too much about it.
It might be a disease called Antracnose. To treat this, it might be rather tricky. Rake away infected leaves and prune infected twigs and leaves from the tree. This should get rid of it next season.
There could be other causes (insects, other kinds of mold, ...).
If you want a reference, you could check out this page right here.
This is normal for this tree and time of year. You are looking at 'honeydew' the urine of aphids, no see ems, or whatever insect is sucking on your tree leaves. Bacteria and fungus become involved also enjoying the carbohydrates this tree made albeit secondary, the excretion is turned black in the process. That black cuts down on photosynthesis so hosing the leaves is good during the growing season.
After the leaves drop clean them up, bag them. I've never seen a weeping birch that wasn't full of aphids, no see ems, ants, spiders yet these insects and goo never harmed the tree.
I would not, however, make a 'sitting area' under or near that tree for humans. I remember trying to prune these things and learning all about no see ems. Ouch.