Effects of mushrooms on their hosts
It probably makes sense to classify mushrooms (fungi) in two separate ways:


symbiotic vs. free-living: symbiotic organisms are those that live in close association with a host (etymologically sym="with", biosis="life"), free-living organisms don't.  So a mycorrhizal fungus that lives within or on the surface of a plant root is a symbiote; a fungus that lives in the soil is free-living.
mutualistic (benefiting from and benefiting a host) vs. parasitic (benefiting from but harming a host) vs. saprotrophic (consuming already-dead material; these are your "decomposers"). (Commensalism [benefiting from, but neither hurting nor harming, a host] is  a fourth possibility.) Mycorrhizal fungi are generally mutualistic. Fungi like chestnut blight or the genus of phythophthora, which includes the organism that causes potato blight, are pathogens; so are the myco-heterotrophs. Fungi such as wood-decay fungi are saprophytic.


To answer your question, species of fungi that are associated with trees may be either mutualistic (e.g. mycorrhizal fungi) or parasitic (e.g. chestnut blight).


Parasitic fungi might also be called pathogenic (i.e., disease-causing)
In popular speech, "symbiotic" is synonymous with "mutualistic" (i.e., organisms that benefit each other); in biology, however, it may refer to either a parasitic or a mutualistic organism. An individual symbiotic organism is often called a "symbiont".
In general, saprotrophs would be free-living (since they don't get their resources from a host, it doesn't really make sense for them to live on a host)
Individuals of a some fungal species can switch lifestyles/categories depending on their environment: these are called facultatively mutualistic/commensal/parasitic, e.g. see Redman et al. 2001.




Redman, Regina S., David D. Dunigan, and Rusty J. Rodriguez. “Fungal Symbiosis from Mutualism to Parasitism: Who Controls the Outcome, Host or Invader?” New Phytologist 151, no. 3 (2001): 705–16. https://doi.org/10.1046/j.0028-646x.2001.00210.x.