TITLE: About the maximum number of leaves adjacent to a vertex in a tree
QUESTION [5 upvotes]: Let $T$ be a finite tree graph with the set of vertices $V(T)$. For an arbitrary vertex $ v \in V(T)$, I define $l(v)$ to be the number of leaves connected to $v$.
In my study, I need to define the following concept:
$$D(T) = \max_{v \in V(T)}l(v). $$
Obviously, $1 \leq D(T) \leq \Delta(T)$, which are achieved by (for example,) the path graphs and the star graphs, respectively. Here, I have two questions:
Question 1. Is there a common notation for $D(T)$?
Question 2. For a random tree $T$, is there some bounds (especially lower bound) for
$$ \dfrac{D(T)}{\vert V(T) \vert}, $$
or, what can one say about the average of the above ratio for the trees of a given order $n$?
Thanks in advance.

REPLY [2 votes]: Expanding on a comment upon OP's request:
To calculate the average over all unlabelled trees on $n$ vertices we can exploit the property that every finite tree has either a centroid or a bicentroid. This allows us to do most of the work with rooted trees. If $R(n, \ell, n')$ denotes the number of rooted trees on $n \ge 1$ vertices with leaf degree no greater than $\ell$ and largest subtree of the root having no more than $n'$ vertices, it's straightforward to observe that $$R(n, \ell, n') = \sum_{\substack{a_1 + 2a_2 + \cdots + n'a_{n'} = n-1 \\ a_i \ge 0, \, a_1 \le \ell}} \prod_i \binom{R(i, \ell, i-1) + a_i - 1}{a_i}$$
and then the number of unrooted trees on $n \ge 1$ vertices with leaf degree no greater than $\ell$ is $$U(n, \ell) = R(n, \ell, \lfloor \tfrac{n-1}{2} \rfloor) + [n \bmod 2 = 0] \binom{R(\tfrac n2, \ell, \tfrac n2-1) + 1}{2}$$ where the first term counts centroidal trees and the second term counts bicentroidal trees.
The average value of $D$ can then be calculated from the sum of $\ell \times (U(n, \ell) - U(n, \ell - 1))$.
$$\begin{array}{lll}n & \textrm{Average value of }D \\
10 & \tfrac{ 273 }{ 106 } &  2.5754716981132075 \\
20 & \tfrac{ 2442856 }{ 823065 } &  2.9679988822267984 \\
30 & \tfrac{ 8019420586 }{ 2471811967 } &  3.2443489606262594 \\
40 & \tfrac{ 139811830066568 }{ 40443361975927 } &  3.4569784319559744 \\
50 & \tfrac{ 19140270453206756713 }{ 5272616851455754767 } &  3.6301273148497755 \\
60 & \tfrac{ 1280181154745986534991397 }{ 339028211512423891688777 } &  3.7760313486450774 \\
70 & \tfrac{ 45682358753537258004625609117 }{ 11707975085109065447981840723 } &  3.901815507930055 \\
80 & \tfrac{ 854579683870799320987268538948428 }{ 212996951362776435118713593401621 } &  4.012168617452553 \\
90 & \tfrac{ 1052333348443216545883200707797525831 }{ 256017690896079594101576682057492020 } &  4.1103930933834185 \\
100 & \tfrac{ 2645869542867297259202984820861933788535463 }{ 630134658347465720563607281977639527019590 } &  4.198895438962389 \\
110 & \tfrac{ 108056089072218514932463235689898358206661706828 }{ 25249910851886927651512825423644528049790467849 } &  4.279464181321791 \\
120 & \tfrac{ 4497776622994307502713890704059983902896242709764300 }{ 1033152520048854877807430091756234770759903219425807 } &  4.353448823588622 \\
130 & \tfrac{ 190236537379552120425249444657539623940185621178617944373 }{ 43021618115267465696869237333175709309993559077318319227 } &  4.421882432916701 \\
140 & \tfrac{ 4078309455388167468312120474298736087566287820459068991699340 }{ 909206958384622679578660502467356659913467956631833840699407 } &  4.485567799254476 \\
150 & \tfrac{ 353859940291181481656671781458954968017223608594806127078280803439 }{ 77854597863245593803382458410206749993828687629640764332291246132 } &  4.545138630254686 \\
160 & \tfrac{ 3101913783575326643405047412687839902474001481340929721745853929687909 }{ 674167392856385067467973832042845361202496256060353873446102172056969 } &  4.601103251868655 \\
170 & \tfrac{ 228641621023362551935592604670190188836671021946402528467487288932247650053 }{ 49129289916495568808202244198612600438761664756803878655268859041109764997 } &  4.65387595489334 \\
180 & \tfrac{ 30578444138206708760484247282898045586908979222565303716833585814401220038148136 }{ 6500796542785541223708685574350747551917347770343511085479226714248864725046787 } &  4.703799593934697 \\
190 & \tfrac{ 1372910069939378604123094002938718076965411651529591781815624762766787704284684359397 }{ 288963010503327141303436810960435570375562139390295662777413482341210574122229958615 } &  4.751161982801847 \\
200 & \tfrac{ 31017734225457597933395862767739806385786293974719248494783415780971891125944778400164787 }{ 6467137160511131588828576867929731880706314086113664851345986440188472113397739601846462 } &  4.796207882346214 \\
210 & \tfrac{ 352410710751392741057157079578720601593230993520260043369781528118511482826645957294214586621 }{ 72824952200684480254538422820789504157985441774873950082120417000877246368175883433716800531 } &  4.839147848394749 \\
220 & \tfrac{ 128797282048478184965954075556518131019553531235842408067945212679480184453626360695959830028141707 }{ 26391994208692151383402347175682684406164602489756559480467014002726855337575585275144893746100626 } &  4.880164834457983 \\
230 & \tfrac{ 5912096543789000433038220579892213157945368680400433791633085366777082281086933708715575870840324133777 }{ 1201787513172706269909923669062213294532536027184453127982144782354836198036307712699780197770445921657 } &  4.919419181000748 \\
240 & \tfrac{ 272563652060306897395747185358655803738754221966330221551036734394657565391873216737292786227592630120780361 }{ 54985024966026897870269075763444229658301263057496157167354274092275126613889793234171034121437216543398430 } &  4.957052437981311 \\
\end{array}
$$