Is there any relationship between the height of a tree and the number of branches emerging from the main stem?
I am particular interested in spruce or fir.
The height is only one of a few factors; also the species and environment are more significant. A spruce growing in unshaded location may have branches to ground level while the same species in a dense woods will only have branches at the top. A southern pine ( in the US) is often 100 ft tall with all the branches in the top 30 ft ; these trees drop the lower branches because of limited light and because various vines climb into the lower branches ( until the branches die and fall).
Not exactly about height-to-branch ratio but an answer more about branching geometry in general.
This is my interpretation of a related question on Reddit, "What determines when/where a branch will grow on a tree?" especially from one answer which I've quoted below. It clarifies a primary mechanism involved. The apical meristem (leader branch / tip of the trunk) releases a hormone, auxin, and as it grows generally upward the concentration of auxin decreases. Too close to the apical meristem and too high auxin, no grow; just the right concentration will generate new branches.
As for the arrangement of hotspots which generate new growth, here's an interesting video on the geometry of plant growth. Not so much about height-to-branch relationship, but it does share important concepts about branch geometry. Basically, branches are forming so as to maximize distance from nearby, previously established branches. All of this seems to be a way to optimize space.
In reality, there are other factors like competition and damage which factor into tree growth. A pristine spruce or fir will have a distinct Phi-like arrangement of branch nodes, as the video linked above describes. How well each branch grows, if at all, will be more complicated.
Basically what you're talking about is phyllotaxy, or the patterning of lateral organs (leaves, branches, etc.) around the axis of growth. Some plants will arrange organs in a spiral around the stem as it grows, or others will alternate from side to side. This varies a lot, but always follows a fairly simple geometric pattern. This will produce buds in specific areas, and is the basis for the patterning of branches that you see in mature trees. The actual biophysical process that arranges this patterning relies on the concept of a morphogen. These are chemicals that can exist in greater or lesser concentrations across a plant tissue. Specifically, we're talking about the distribution of auxin across the surface of the shoot apical meristem. The details are quite complex (and the details super controversial!) but suffice it to say that the plant cells transport auxin to neighboring cells based on feedback loops that produce spots of high auxin concentration. The spots with high auxin develop into lateral organs (leaves). You can model with computers and recapitulate observed phyllotaxy seen in nature.
Now, I'm sure you've noticed that trees don't have well-patterned branches. Still, this very well organized structure is the basis for plant architecture. Look at young tissue or flowers, and you'll easily see these patterns. But back to branches.
When a lateral organ is formed, it is accompanied with an axillary meristem and a segment of stem. This basic structural unit of plants is called a phytomer. Now, the axillary meristem can do a few things. For trees, it can produce leaves, a flower, and/or make a branch. The 'algorithm' that determines what happens will vary from species to species and depend on things like apical dominance and environmental factors (e.g exposure to light in that area). Some of these axillary meristems will become branch meristems, and this is somewhat random from a patterning perspective. This is why most trees don't follow strict patterns. Over time, some branches will grow at different rates, others will terminate, and successive branching decisions will determine it's overall form.
Most trees can also form meristems de novo from tissues. If you've ever seen a forest tree suddenly exposed to light (from a neighbor falling down) you'll know what I'm talking about, as the trunk might be covered in leaves. I haven't ever studied this, but I presume that this process is quite random, but I also don't think that it's a major influence on tree architecture.
via Reddit user Sluisifer